1 Datos

Consideremos de nuevo los datos de tipos de cambio que hemos venido empleando en el curso.

data <- read.csv("est46114_s06_data.csv")
data <- as.data.frame(data)
data <- as.matrix(data)
T <- nrow(data); p <- ncol(data)-2
datats <- ts(data[,c(3,4,p+2)],start=c(1970, 1), end=c(2010, 12), frequency=12)
plot(datats)

2 Modelo de factores

El modelo de factores considera que tiempo-a-tiempo (i.e. para cada \(t\) mes), el bloque de los \(80\) tipos de cambio, o de los \(3\) tipos de cambio (caso reducido para ‘Canada’,‘Mexico’,‘Zambia’), denotados por \(\boldsymbol{X}_t\), pueden modelarse como \[ \boldsymbol{X}_t \sim \text{N}_p(\boldsymbol{x}|\boldsymbol{\theta},\boldsymbol{\Omega}), \] donde

  • \(\boldsymbol{\theta}\) es la media general en \(t\), y

  • \(\boldsymbol{\Omega}\) es la matriz de varianzas y covarianzas de los \(p\) tipos de cambio.

2.1 Supuestos

Asi, el modelo implicitamente supone:

  1. Homogeneidad, i.e. $ _p(|,)$ en su ‘forma estructural’ y ‘parametros’ sea el mismo para todo \(t\).

  2. Invarainza al orden, i.e. \[ \mathbb{P}(\boldsymbol{x}_1,\ldots,\boldsymbol{x}_t)=\mathbb{P}(\boldsymbol{x}_{\sigma(1)},\ldots,\boldsymbol{x}_{\sigma(t)}). \] Induciendo que el ‘modelo lea’ las observaciones independientemente o condicionalmente independiente en \(t\).

2.2 Consideraciones

Hemos visto que los supuestos (1) y (2) no son sustentables en este conjunto de datos.

2.2.1 Independencia

Si los datos tuviesen proveyeran informacion acerca de la independencia en \(t\), tendriamos que ‘muestralmente’, las correlaciones de los datos en \(t\) respecto a tiempos previos (rezagos), seria ‘nula’.

La funcion de autocorrelacion nos brinda una herramienta para inspeccionar este supuesto.

acf(datats[,"Canada"])

acf(datats[,"Mexico"])

acf(datats[,"Zambia"])

2.2.2 Tendencia

Si el modelo homogeneo en tendencia (medias, o \(\boldsymbol{\theta}_t\)s) fuese admisible para el conjunto de datos, tendriamos que para todo orden de observaciones el comportamiento de las medias por ‘subsegmentos’ de datos seria muy semejante. En la grafica vemos que en este caso, esto no se cumple.

datats <- ts(data[,3:(p+2)],start=c(1970, 1), end=c(2010, 12), frequency=12)
Y <- T/12
years <- seq(1970,2010,1)
meansts <- NA * datats[c(1:41),]
t <- 1
for(t in 1:Y){
  meansts[t, ] <- colMeans(data[which(data[,"Year"]==years[t]),3:(p+2)])
}
plot(meansts[,"Canada"])

plot(meansts[,"Mexico"])

plot(meansts[,"Zambia"])

rep.row<-function(x,n){
   matrix(rep(x,each=n),nrow=n)
}
rep.col<-function(x,n){
   matrix(rep(x,each=n), ncol=n, byrow=TRUE)
}

Corrigiendo por medias temporales, i.e. relajando el supesto que \[ \mathbb{E}(\boldsymbol{X}_t)=\boldsymbol{\theta}, \] para todo \(t\), por el siguiente, \[ \mathbb{E}(\boldsymbol{X}_t)=\boldsymbol{\theta}_a, \] donde \(a\) corresponde al anio calendario correspondiente a cada \(t\), observamos un efecto de correccion en tendencia.

datatsc <- datats
t <- 1
for(t in 1:Y){
  datatsc[which(data[,"Year"]==years[t]),] <- datats[which(data[,"Year"]==years[t]),] - rep.row(meansts[t, ],12)
}
plot(datatsc[,c("Canada","Mexico","Zambia")])

2.2.3 Independencia (revisada)

El efecto de correccion en tendencia implica necesariamente un efecto en terminos de la estructura de dependencia intrinseca (autocorrelacion).

acf(datatsc[,"Canada"])

acf(datatsc[,"Mexico"])

acf(datatsc[,"Zambia"])

2.3 Invarianza al orden

if(!require("gtools")){install.packages("gtools")}
## Loading required package: gtools
library("gtools")

t.index <- c(1:T)

set.seed(1)

t.sample1 <- sample(t.index)

t.sample2 <- sample(t.index)

t.sample3 <- sample(t.index)

2.3.0.1 Datos originales

plot(ts(datatsc[t.index,c("Canada","Mexico","Zambia")],frequency=12))

#### Primera permutacion

plot(ts(datatsc[t.sample1,c("Canada","Mexico","Zambia")],frequency=12))

#### Segunda permutacion

plot(ts(datatsc[t.sample2,c("Canada","Mexico","Zambia")],frequency=12))

2.3.0.2 Tercera permutacion

plot(ts(datatsc[t.sample3,c("Canada","Mexico","Zambia")],frequency=12))

2.4 Implementacion

if(!require('MCMCpack')){install.packages("MCMCpack")}
## Loading required package: MCMCpack
## Loading required package: coda
## Loading required package: MASS
## ##
## ## Markov Chain Monte Carlo Package (MCMCpack)
## ## Copyright (C) 2003-2019 Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park
## ##
## ## Support provided by the U.S. National Science Foundation
## ## (Grants SES-0350646 and SES-0350613)
## ##
## 
## Attaching package: 'MCMCpack'
## The following objects are masked from 'package:gtools':
## 
##     ddirichlet, rdirichlet
library("MCMCpack")

datatsc.3 <- as.data.frame(datatsc[t.index,c("Canada","Mexico","Zambia")])

fa.posterior <- MCMCfactanal(~Canada+Mexico+Zambia, 
                             factors=1,
                             lambda.constraints=list(Mexico1=c(1,0), 
                                                     Zambia1=c(1,0)),
                             verbose=0, store.scores=FALSE, 
                             a0=1, b0=0.15,
                             data=datatsc.3, 
                             burnin=50, mcmc=150, thin=1)

fa.posterior
## Markov Chain Monte Carlo (MCMC) output:
## Start = 51 
## End = 200 
## Thinning interval = 1 
##        LambdaCanada_1 LambdaMexico_1 LambdaZambia_1 PsiCanada PsiMexico
##   [1,]  -1.313440e-01     -0.8038129     -0.2816096 1.0061454 0.4116594
##   [2,]  -1.570016e-01     -0.7579327     -0.3039861 0.9361589 0.3959228
##   [3,]  -1.614056e-01     -0.7879895     -0.2256421 0.8798418 0.3875877
##   [4,]  -1.680800e-01     -0.8266690     -0.1423309 0.8771918 0.3497977
##   [5,]  -1.674268e-01     -0.8235891     -0.1972571 0.9725896 0.3506025
##   [6,]  -7.691657e-02     -0.7753728     -0.2506273 0.9330723 0.2983826
##   [7,]  -7.955401e-02     -0.8268506     -0.3130412 0.8937084 0.3833808
##   [8,]   1.069530e-02     -0.7540654     -0.3570257 1.0461590 0.3153575
##   [9,]  -2.546568e-02     -0.7507844     -0.3059460 1.0892251 0.4065127
##  [10,]   1.595217e-03     -0.7802823     -0.2139933 1.0323348 0.4558982
##  [11,]  -1.476444e-01     -0.7833344     -0.2675254 0.9937036 0.3797556
##  [12,]  -1.806595e-01     -0.7563406     -0.2508719 0.9270239 0.3797318
##  [13,]  -1.002755e-01     -0.7602890     -0.2597754 0.9555030 0.4363572
##  [14,]  -5.932690e-02     -0.7225603     -0.2714403 0.9803167 0.5128466
##  [15,]  -1.517905e-01     -0.7110268     -0.3177422 0.9602993 0.4525728
##  [16,]  -1.135719e-01     -0.6442257     -0.3620564 0.9689639 0.5089521
##  [17,]  -1.688329e-01     -0.6683868     -0.3208790 1.0387238 0.5797079
##  [18,]  -2.325099e-01     -0.6685861     -0.2922009 1.0936516 0.6332812
##  [19,]  -2.388811e-01     -0.6050263     -0.2730728 0.9622687 0.5713836
##  [20,]  -2.368702e-01     -0.6232182     -0.2488101 0.9746449 0.6138756
##  [21,]  -1.724804e-01     -0.5906095     -0.1687467 0.8820698 0.5279362
##  [22,]  -1.059608e-01     -0.6103843     -0.2437824 1.0526378 0.5774108
##  [23,]  -1.081365e-01     -0.6252663     -0.2151890 1.0429392 0.6938378
##  [24,]  -2.885507e-01     -0.5991988     -0.2432485 0.9378205 0.6576875
##  [25,]  -3.032972e-01     -0.5908190     -0.2970895 1.0386433 0.6281160
##  [26,]  -2.357407e-01     -0.6287761     -0.3321617 0.9129389 0.6336322
##  [27,]  -1.749892e-01     -0.6327618     -0.3414023 0.8865521 0.5184952
##  [28,]  -1.306157e-01     -0.5769780     -0.3203061 0.9416544 0.6357294
##  [29,]  -1.006285e-01     -0.4986698     -0.3574179 0.9608514 0.6208821
##  [30,]  -2.191599e-01     -0.5786390     -0.3617068 0.9594067 0.7527005
##  [31,]  -2.697413e-01     -0.5614428     -0.3557484 0.9750482 0.7097464
##  [32,]  -1.994997e-01     -0.5647001     -0.3617366 0.8070104 0.7523981
##  [33,]  -1.866581e-01     -0.6063550     -0.3755794 0.9279500 0.8036021
##  [34,]  -1.901035e-01     -0.5919677     -0.2686846 0.8962285 0.7149832
##  [35,]  -1.002256e-01     -0.6099510     -0.3414800 0.9886101 0.6434619
##  [36,]  -2.192024e-01     -0.6341711     -0.2659336 1.0018529 0.6627565
##  [37,]  -1.969789e-01     -0.5223273     -0.3100717 0.8675846 0.6441029
##  [38,]  -2.074480e-01     -0.5225189     -0.4567816 0.9437811 0.7864313
##  [39,]  -1.005176e-01     -0.5804788     -0.3778731 1.0843592 0.7024209
##  [40,]  -1.349020e-01     -0.6031803     -0.3822221 1.0054097 0.6922420
##  [41,]  -9.354188e-02     -0.6073132     -0.2473797 1.0390408 0.6091122
##  [42,]  -3.712989e-02     -0.6624150     -0.2607910 0.9365328 0.6212531
##  [43,]  -1.428434e-01     -0.6142774     -0.2821468 0.9547870 0.5463931
##  [44,]  -8.660184e-02     -0.6400922     -0.2701131 1.0106911 0.5063704
##  [45,]  -5.362241e-02     -0.6871810     -0.2958875 0.9190562 0.5746872
##  [46,]  -6.375656e-02     -0.7577042     -0.3497381 0.8958022 0.5283322
##  [47,]  -1.381138e-01     -0.7947216     -0.3055979 0.9915012 0.5127269
##  [48,]  -1.125345e-01     -0.7608732     -0.2967274 1.0020842 0.5473486
##  [49,]  -7.343879e-02     -0.6733398     -0.3029775 1.0499278 0.4710796
##  [50,]  -1.737512e-01     -0.7025193     -0.3215639 0.9840202 0.5755468
##  [51,]  -1.559548e-01     -0.6834554     -0.3278621 0.9401004 0.6172367
##  [52,]  -1.899308e-01     -0.6561680     -0.2766643 0.8863264 0.5176300
##  [53,]  -1.609041e-01     -0.6287876     -0.3489765 0.9936052 0.6477754
##  [54,]  -1.658593e-01     -0.6293869     -0.3423813 0.9302969 0.5825986
##  [55,]  -3.077530e-01     -0.6333443     -0.2567675 0.9490136 0.6015178
##  [56,]  -2.511484e-01     -0.7089937     -0.3093567 0.9602045 0.5831954
##  [57,]  -1.112871e-01     -0.6566115     -0.2281089 0.8235235 0.5349589
##  [58,]  -1.792509e-01     -0.6984216     -0.2313344 1.1014390 0.5172283
##  [59,]  -3.945610e-02     -0.7244162     -0.2582507 0.9326448 0.5051566
##  [60,]  -5.749231e-06     -0.7622402     -0.2763557 1.0622995 0.5665040
##  [61,]  -4.099813e-02     -0.8475316     -0.3182463 0.9974873 0.5262669
##  [62,]  -1.103989e-01     -0.7801294     -0.3307596 1.0088856 0.4252963
##  [63,]  -1.260015e-01     -0.7434451     -0.2884836 0.8865973 0.3748638
##  [64,]  -7.381889e-02     -0.7695053     -0.2492356 0.9519695 0.4074981
##  [65,]  -1.081957e-01     -0.7135481     -0.2976787 0.8576031 0.3767433
##  [66,]  -1.017797e-01     -0.7634943     -0.3130865 1.1135997 0.3936456
##  [67,]  -1.021198e-01     -0.7766113     -0.2325455 0.9270748 0.3795737
##  [68,]  -8.478626e-02     -0.8112321     -0.2204310 0.8834349 0.4068485
##  [69,]  -1.404440e-01     -0.8109807     -0.2271326 0.9272189 0.3739250
##  [70,]  -1.936729e-01     -0.7515774     -0.2143876 0.9148333 0.4229171
##  [71,]  -1.394633e-01     -0.7484795     -0.2287667 1.0004243 0.3903694
##  [72,]  -1.286486e-01     -0.7308400     -0.2279740 0.9935352 0.3219727
##  [73,]  -9.846330e-02     -0.8012564     -0.2354420 1.0648581 0.3581573
##  [74,]  -1.193247e-01     -0.8148217     -0.2446225 0.9996250 0.3333158
##  [75,]  -1.304124e-01     -0.8367927     -0.2134898 0.8913346 0.3896981
##  [76,]  -1.345060e-01     -0.8240153     -0.3176188 1.0182160 0.3270810
##  [77,]  -7.990427e-02     -0.8041979     -0.2318064 1.0237494 0.3333583
##  [78,]  -3.214348e-02     -0.7473483     -0.1818638 0.9971271 0.3416685
##  [79,]  -3.892430e-02     -0.7474312     -0.2412871 1.0150209 0.3552439
##  [80,]  -6.789114e-02     -0.7950808     -0.2188696 1.0157041 0.3737743
##  [81,]  -1.614006e-01     -0.7999718     -0.3323355 1.0065656 0.3851088
##  [82,]  -1.824901e-01     -0.8249488     -0.2505241 0.9943472 0.3582281
##  [83,]  -2.363117e-01     -0.7787434     -0.2864547 0.8602734 0.3320010
##  [84,]  -2.052754e-01     -0.7834870     -0.2810964 0.9245595 0.3101502
##  [85,]  -2.805652e-01     -0.7982204     -0.1859628 0.9487280 0.3037625
##  [86,]  -1.939421e-01     -0.8271568     -0.1790840 1.0709482 0.3301610
##  [87,]  -1.505755e-01     -0.8371194     -0.1631539 1.0643666 0.3202306
##  [88,]  -9.323318e-02     -0.8565474     -0.2222155 0.9827947 0.2780590
##  [89,]  -6.633465e-02     -0.7974952     -0.2741091 0.9688497 0.2907187
##  [90,]  -1.248921e-01     -0.8612397     -0.2579029 0.9606209 0.2966333
##  [91,]  -8.603955e-02     -0.8629897     -0.2734082 1.0815317 0.2954396
##  [92,]  -1.895134e-01     -0.8982412     -0.2731863 1.0317103 0.2357103
##  [93,]  -1.444651e-01     -0.8766824     -0.2521674 1.1019412 0.2258737
##  [94,]  -1.804944e-01     -0.8357706     -0.1931679 0.9955691 0.2391250
##  [95,]  -1.300700e-01     -0.8841026     -0.2028720 0.8875567 0.2610720
##  [96,]  -4.155302e-02     -0.8638341     -0.2932436 0.9069720 0.2214591
##  [97,]  -1.066283e-01     -0.8934176     -0.2637394 1.0435918 0.2505003
##  [98,]  -2.108477e-02     -0.8396800     -0.2967805 0.9771144 0.2549567
##  [99,]  -1.189242e-01     -0.8269176     -0.2348267 0.9379920 0.2538293
## [100,]  -1.531149e-01     -0.8288699     -0.1736272 0.9897010 0.2729557
## [101,]  -7.343503e-02     -0.8586443     -0.2671735 1.0584125 0.2488753
## [102,]  -1.262218e-01     -0.8917072     -0.2734164 0.9193474 0.2667662
## [103,]  -2.064438e-01     -0.9166786     -0.2580935 0.9495710 0.2312843
## [104,]  -1.302428e-01     -0.8724548     -0.2095246 1.0573440 0.2623110
## [105,]  -1.774327e-01     -0.8887345     -0.2148600 0.9689191 0.2439879
## [106,]  -2.604726e-01     -0.9080614     -0.2259497 1.0713583 0.2203756
## [107,]  -1.574909e-01     -0.8934616     -0.2749616 0.9571631 0.2366237
## [108,]  -1.945140e-01     -0.9024526     -0.2453184 0.9332897 0.2784568
## [109,]  -1.966169e-01     -0.9440960     -0.2863667 0.9654002 0.3109694
## [110,]  -4.704324e-02     -0.9318366     -0.2786000 0.9367688 0.3156816
## [111,]  -1.510234e-01     -0.8640621     -0.2818296 1.0298093 0.2734327
## [112,]  -1.154050e-01     -0.8722377     -0.1393866 1.0449739 0.2562684
## [113,]  -1.126535e-01     -0.9027718     -0.2592732 0.9233598 0.2533854
## [114,]  -1.189043e-01     -0.8558491     -0.2594629 1.0682758 0.2248855
## [115,]  -1.289682e-01     -0.8030648     -0.2982030 0.9807193 0.2543455
## [116,]  -7.538957e-02     -0.8541609     -0.2541948 1.1411476 0.2497804
## [117,]  -1.041427e-02     -0.8687691     -0.2238398 0.9765312 0.2787748
## [118,]  -2.440391e-02     -0.8787642     -0.1779331 1.0956801 0.2317387
## [119,]  -1.243644e-01     -0.8421912     -0.2167075 0.9727927 0.2603127
## [120,]  -5.717118e-02     -0.8603274     -0.2114056 0.9788596 0.2498312
## [121,]  -4.642273e-02     -0.8931270     -0.2393605 0.9179936 0.2508926
## [122,]  -6.934878e-02     -0.8494479     -0.2084370 0.9394036 0.2717849
## [123,]  -5.017460e-02     -0.8332300     -0.1863026 1.0140950 0.2674003
## [124,]  -2.034862e-01     -0.8238387     -0.1837118 0.9704464 0.2672748
## [125,]  -1.696534e-01     -0.8888636     -0.2731107 1.1119611 0.3077279
## [126,]  -2.159597e-01     -0.8826986     -0.3340160 0.8457513 0.3167757
## [127,]  -1.573342e-01     -0.8559097     -0.3328415 0.9603885 0.3021574
## [128,]  -1.069889e-01     -0.8331950     -0.3130323 1.0457178 0.2934011
## [129,]  -1.516062e-01     -0.8678489     -0.3002747 0.9778591 0.2754519
## [130,]  -1.066895e-01     -0.9016105     -0.2985903 0.9170580 0.2756804
## [131,]  -1.533520e-01     -0.9096727     -0.2326407 1.1001621 0.2728300
## [132,]  -1.115418e-01     -0.9191171     -0.2250652 0.9614081 0.2295588
## [133,]  -1.420211e-01     -0.8818978     -0.3099841 0.9816122 0.2254461
## [134,]  -1.594246e-01     -0.8516391     -0.2522270 0.9899600 0.2271001
## [135,]  -8.371003e-02     -0.8769231     -0.2455377 0.9558094 0.2105616
## [136,]  -8.115337e-02     -0.8952963     -0.2760660 0.9957728 0.2121907
## [137,]  -2.610045e-01     -0.9152883     -0.3146934 0.9267679 0.2515472
## [138,]  -1.224313e-01     -0.8735643     -0.2451977 0.9661833 0.2663440
## [139,]  -1.948237e-01     -0.9253350     -0.2978792 0.9868680 0.2542990
## [140,]  -1.645429e-01     -0.9240053     -0.2841008 0.9279769 0.2860417
## [141,]  -1.165529e-01     -0.8783499     -0.1670780 1.1467915 0.3062565
## [142,]  -2.785001e-01     -0.8313462     -0.1818295 1.0068730 0.2975372
## [143,]  -1.868471e-01     -0.8444385     -0.3091613 0.9361075 0.2972175
## [144,]  -3.257599e-02     -0.8621752     -0.3017831 0.9548435 0.2741142
## [145,]  -1.025659e-01     -0.8801877     -0.2387719 0.9168576 0.2615302
## [146,]  -6.654101e-02     -0.8737189     -0.2421176 0.9381385 0.2720359
## [147,]  -1.790850e-01     -0.8413226     -0.2090160 1.1239113 0.2752455
## [148,]  -1.522042e-01     -0.8109911     -0.2447516 1.0094764 0.3120314
## [149,]  -9.590794e-02     -0.8449021     -0.1921708 0.9690455 0.2825912
## [150,]  -1.329869e-01     -0.8125215     -0.1301149 0.9733932 0.2621129
##        PsiZambia
##   [1,] 0.9954873
##   [2,] 1.0070362
##   [3,] 1.0516263
##   [4,] 0.9532203
##   [5,] 0.9044722
##   [6,] 0.9326477
##   [7,] 0.8615549
##   [8,] 0.9336869
##   [9,] 0.9683080
##  [10,] 1.0592374
##  [11,] 1.0108144
##  [12,] 1.0153966
##  [13,] 0.9147160
##  [14,] 0.9379959
##  [15,] 0.9355091
##  [16,] 0.8692317
##  [17,] 0.8701564
##  [18,] 0.8482479
##  [19,] 0.9563067
##  [20,] 0.9721528
##  [21,] 0.8854596
##  [22,] 0.8447454
##  [23,] 0.9042113
##  [24,] 0.8544734
##  [25,] 1.0202484
##  [26,] 0.9032024
##  [27,] 0.8456380
##  [28,] 0.8577884
##  [29,] 0.8353255
##  [30,] 0.8546962
##  [31,] 0.8581595
##  [32,] 0.9264553
##  [33,] 0.9564432
##  [34,] 0.8896792
##  [35,] 0.8893527
##  [36,] 0.8444730
##  [37,] 0.8553654
##  [38,] 0.8178582
##  [39,] 0.8688949
##  [40,] 0.8919509
##  [41,] 0.8775137
##  [42,] 1.0250008
##  [43,] 0.8426380
##  [44,] 0.9157439
##  [45,] 0.8800744
##  [46,] 0.8549704
##  [47,] 0.8669941
##  [48,] 0.9916016
##  [49,] 0.8879934
##  [50,] 0.8706642
##  [51,] 0.8728135
##  [52,] 0.8586143
##  [53,] 0.9154176
##  [54,] 0.8964398
##  [55,] 0.9295307
##  [56,] 0.8601873
##  [57,] 0.7869318
##  [58,] 0.9410080
##  [59,] 0.8597775
##  [60,] 0.9493921
##  [61,] 0.9446040
##  [62,] 0.9469455
##  [63,] 0.9537974
##  [64,] 0.8690808
##  [65,] 0.8848051
##  [66,] 0.9601448
##  [67,] 0.8527361
##  [68,] 1.0435621
##  [69,] 0.9884363
##  [70,] 0.9397550
##  [71,] 0.8993054
##  [72,] 1.0605649
##  [73,] 0.8918531
##  [74,] 0.8743727
##  [75,] 1.0309277
##  [76,] 0.9828107
##  [77,] 0.9252888
##  [78,] 0.9770418
##  [79,] 0.9028271
##  [80,] 0.9302682
##  [81,] 0.9615108
##  [82,] 0.9258265
##  [83,] 0.9225817
##  [84,] 0.8744992
##  [85,] 1.0142766
##  [86,] 1.0306481
##  [87,] 1.0891932
##  [88,] 0.9930969
##  [89,] 1.1160303
##  [90,] 1.0070212
##  [91,] 0.9201742
##  [92,] 0.8833454
##  [93,] 0.9885911
##  [94,] 1.0343708
##  [95,] 0.8939333
##  [96,] 1.0102840
##  [97,] 0.8639778
##  [98,] 0.8783241
##  [99,] 0.9714797
## [100,] 0.8965222
## [101,] 0.8833697
## [102,] 0.9301872
## [103,] 0.9453331
## [104,] 0.8468230
## [105,] 0.9617060
## [106,] 0.9218789
## [107,] 0.9446461
## [108,] 1.0103551
## [109,] 0.9514989
## [110,] 0.9294979
## [111,] 0.9604210
## [112,] 0.9589084
## [113,] 0.9151665
## [114,] 0.9713864
## [115,] 0.9745754
## [116,] 0.9635867
## [117,] 1.1077937
## [118,] 0.9674457
## [119,] 1.0150687
## [120,] 0.9676572
## [121,] 0.8978634
## [122,] 0.8401712
## [123,] 0.8874293
## [124,] 1.0031673
## [125,] 0.9510003
## [126,] 0.8455523
## [127,] 0.9084703
## [128,] 0.9190738
## [129,] 0.9280782
## [130,] 0.9333786
## [131,] 1.0343486
## [132,] 0.9162654
## [133,] 1.0132667
## [134,] 0.9907024
## [135,] 1.0044713
## [136,] 1.0375815
## [137,] 0.9628848
## [138,] 0.9025556
## [139,] 1.0078061
## [140,] 0.8685727
## [141,] 1.0256659
## [142,] 0.9123188
## [143,] 0.9378514
## [144,] 0.8856493
## [145,] 0.9027066
## [146,] 0.9548190
## [147,] 1.0734965
## [148,] 0.9383125
## [149,] 1.0101698
## [150,] 1.0216437
## attr(,"constraints")
## attr(,"constraints")$Mexico1
## [1] 1 0
## 
## attr(,"constraints")$Zambia1
## [1] 1 0
## 
## attr(,"n.manifest")
## [1] 3
## attr(,"n.factors")
## [1] 1
summary(fa.posterior)
## 
## Iterations = 51:200
## Thinning interval = 1 
## Number of chains = 1 
## Sample size per chain = 150 
## 
## 1. Empirical mean and standard deviation for each variable,
##    plus standard error of the mean:
## 
##                   Mean      SD Naive SE Time-series SE
## LambdaCanada_1 -0.1351 0.06616 0.005402       0.009056
## LambdaMexico_1 -0.7755 0.10815 0.008830       0.065804
## LambdaZambia_1 -0.2665 0.05541 0.004524       0.008186
## PsiCanada       0.9781 0.06602 0.005390       0.005390
## PsiMexico       0.3995 0.15549 0.012695       0.108898
## PsiZambia       0.9351 0.06511 0.005316       0.008709
## 
## 2. Quantiles for each variable:
## 
##                   2.5%     25%     50%      75%    97.5%
## LambdaCanada_1 -0.2791 -0.1787 -0.1305 -0.09331 -0.01815
## LambdaMexico_1 -0.9205 -0.8619 -0.8034 -0.69945 -0.56380
## LambdaZambia_1 -0.3658 -0.3037 -0.2666 -0.22827 -0.16600
## PsiCanada       0.8595  0.9340  0.9731  1.01479  1.11241
## PsiMexico       0.2239  0.2722  0.3457  0.52432  0.72527
## PsiZambia       0.8420  0.8837  0.9299  0.97643  1.06412

2.4.0.1 Comentarios sobre la implementacion

  • Estamos considerando que los niveles medios \((\boldsymbol{\theta}_a)_{a\geq 1970}\) son “fijos”/“conocidos”para el modelo, aunque estos los hayamos calculado con los datos.

  • Los coeficientes (LambdaCanada_1,LambdaMexico_1,LambdaZambia_1) corresponden al vector de cargas (unico, en este caso) contenido en la matriz \(\Lambda\).

  • Los coeficientes (PisCanada,PsiMexico,PsiZambia) corresponden a las varianzas contenidas en la diagonal de la matriz \(\Sigma\).

2.5 Diagrama de dependencia

Diagrama de dependencia del modelo de factores estatico

Diagrama de dependencia del modelo de factores estatico

3 Especificacion dinamica en factores

En este caso, es de interes recuperar informacion respecto a la tendencia observada en los datos \((\boldsymbol{X}_t)_{t\geq 1}\) en terminos de los factores latentes, \((\boldsymbol{f}_t)_{t\geq 1}\) y de un componente en niveles \((\boldsymbol{\theta}_t)_{t\geq 1}\).

Asi, para los datos \((p\times 1)\)-dimensionales, el modelo es de la forma \[\begin{eqnarray} \boldsymbol{X}_t|\boldsymbol{\theta}_t,\boldsymbol{f}_t & \sim & N_p(\boldsymbol{x}|\boldsymbol{\theta}_t+\Lambda \boldsymbol{f}_t,\Sigma)\nonumber \\ \end{eqnarray}\] para \(t\geq 1\), donde \(\boldsymbol{\theta}_t\) es un vector \((p\times 1)\)-dimensional, \(\boldsymbol{f}_t\) es un vector \((k\times 1)\)-dimensional (con \(k<<p\)), tales que \[\begin{eqnarray} \boldsymbol{\theta}_t|\boldsymbol{\theta}_{t-1} & \sim & N_p(\boldsymbol{\theta}|\boldsymbol{f}_{t-1},\boldsymbol{W}) \nonumber, \\ \boldsymbol{f}_t|\boldsymbol{f}_{t-1} & \sim & N_k(\boldsymbol{f}|\Lambda \boldsymbol{f}{t-1},\boldsymbol{G}) \nonumber, \end{eqnarray}\] donde

  • \(\Lambda\) es la matriz \((p\times k)\)-dimensional de cargas de factores,

  • \(\boldsymbol{W}\) es una matriz de varianzas y covarianzas (tipicamente conocida o preespecificada) de dimension \((p\times p)\) y

  • \(\boldsymbol{G}\) es una matriz diagonal de varianzas de dimension \((k\times k)\) (tambien, usualmente, fija o predefinida).

3.1 Parametros

Los parametros del modelo quedan entonces especificados como \[ \left(\Lambda,\boldsymbol{\Sigma}\right), \] siendo las variables latentes \[ \left((\boldsymbol{f}_{t})_{t\geq 1}\right). \] Las condiciones de identificabilidad del modelo son iguales a las tradicionales, respecto a \(\Lambda\).

3.2 Observaciones

  • El parametro \(\boldsymbol{\theta}_t\) brinda una lectura acerca de la tendencia de los datos, en \(t\), de manera separada para las \(p\) dimensiones.

  • El parametro \(\boldsymbol{f}_t|\boldsymbol{x}_t\) brinda informacion acerca del componente coyuntural de correlacion subyacente a \(\boldsymbol{x}_t\) en \(t\).

  • El componente de tendencia coyuntural agregado es, en este caso, \[\begin{eqnarray} \mathbb{E}(\boldsymbol{x}|\boldsymbol{f}_t) & = & \boldsymbol{\theta}_t + \Lambda \boldsymbol{f}_t, \nonumber \end{eqnarray}\] mientras que el compoente de tendencia coyuntural marginalizado es, \[\begin{eqnarray} \mathbb{E}(\boldsymbol{x}|\boldsymbol{f}_{t-1}) & = & \int \boldsymbol{x}_t N_p(\boldsymbol{x}_t|\boldsymbol{\theta}_t+\Lambda\boldsymbol{t}_t)N_k(\boldsymbol{f}_t|\boldsymbol{f}_{t-1},\boldsymbol{G})\text{d}\boldsymbol{f}_t \nonumber \\ & = & \boldsymbol{\theta}_t + \Lambda \boldsymbol{f}_{t-1}. \nonumber \end{eqnarray}\]

Esta ultima es una media que hereda un componente del periodo previo. A diferencia del modelo estatico original, el efecto de \(\boldsymbol{f}_t\) no se desvanece necesariamente.

3.3 Asociaciones

El modelo anterior puede verse como un caso particular del modelo state-space o dynamic linear model, donde el componente de estados es dual, i.e., \[ (\boldsymbol{\theta}_t,\boldsymbol{f}_t)_{t\geq 1}. \] Vean la referencia de West & Harrison en las lecturas complementarias.

Es de notar que el componente de reduccion de dimensionalidad incorporado induce a que la estimacion de parametros no pueda realizarse de manera analitica cerrada empleando el filtro de Kalman. En este caso, algorimos del tipo MCMC y relacionados son empleados.

3.4 Implementacion

if(!require("bayesdfa")){install.packages("bayesdfa")}
## Loading required package: bayesdfa
## Loading required package: Rcpp
library("bayesdfa")

dfa.posterior <- fit_dfa(y = t(datatsc.3), iter = 150, chains = 1)
## trying deprecated constructor; please alert package maintainer
## 
## SAMPLING FOR MODEL 'dfa' NOW (CHAIN 1).
## Chain 1: 
## Chain 1: Gradient evaluation took 0.000451 seconds
## Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 4.51 seconds.
## Chain 1: Adjust your expectations accordingly!
## Chain 1: 
## Chain 1: 
## Chain 1: WARNING: There aren't enough warmup iterations to fit the
## Chain 1:          three stages of adaptation as currently configured.
## Chain 1:          Reducing each adaptation stage to 15%/75%/10% of
## Chain 1:          the given number of warmup iterations:
## Chain 1:            init_buffer = 11
## Chain 1:            adapt_window = 57
## Chain 1:            term_buffer = 7
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## Chain 1: 
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## Chain 1:                23.484 seconds (Sampling)
## Chain 1:                57.373 seconds (Total)
## Chain 1:
## Warning in sqrt(ess): NaNs produced
## Inference for the input samples (1 chains: each with iter=75; warmup=37):
## 
##                  mean se_mean   sd    2.5%     25%     50%     75%   97.5%
## x[1,1]            0.3     0.2  1.3    -1.9    -0.5     0.3     0.8     2.8
## x[1,2]            0.6     0.2  1.5    -2.1    -0.6     0.6     1.3     3.3
## x[1,3]            0.6     0.2  1.3    -1.5    -0.2     0.5     1.4     3.1
## x[1,4]            0.6     0.1  0.9    -1.1     0.0     0.6     1.1     2.2
## x[1,5]            0.4     0.1  1.0    -1.0     0.0     0.4     0.8     2.4
## x[1,6]           -0.1     0.2  1.2    -2.5    -1.0    -0.1     0.7     2.0
## x[1,7]           -0.4     0.0  1.6    -4.1    -0.9    -0.2     0.7     1.8
## x[1,8]           -0.8     0.2  1.4    -3.6    -1.5    -0.6     0.1     1.3
## x[1,9]           -1.0     0.2  1.4    -3.7    -1.9    -1.1    -0.2     1.3
## x[1,10]          -0.7     0.1  1.2    -3.1    -1.3    -0.8     0.2     1.3
## x[1,11]          -0.4     0.2  1.1    -2.2    -1.2    -0.5     0.4     1.7
## x[1,12]          -0.2     0.2  1.1    -2.1    -1.0    -0.2     0.5     1.6
## x[1,13]           0.6     0.2  1.7    -1.8    -0.7     0.3     1.4     3.7
## x[1,14]           0.3     0.2  1.4    -1.5    -0.5     0.0     1.1     3.2
## x[1,15]           0.4     0.1  1.5    -2.2    -0.4     0.2     1.4     2.9
## x[1,16]           0.1     0.2  1.2    -2.1    -0.6     0.0     1.0     2.1
## x[1,17]           0.1     0.1  1.1    -1.5    -0.6     0.0     0.7     2.3
## x[1,18]           0.3     0.2  1.2    -2.4    -0.4     0.3     1.2     2.4
## x[1,19]          -0.1     0.2  1.7    -3.8    -1.0     0.3     1.0     2.4
## x[1,20]          -0.1     0.2  1.2    -2.2    -0.5    -0.1     0.5     2.4
## x[1,21]          -0.1     0.2  1.2    -2.1    -0.9    -0.2     0.9     1.8
## x[1,22]          -0.1     0.1  1.2    -1.8    -1.0    -0.3     0.5     2.6
## x[1,23]           0.2     0.2  1.8    -2.0    -1.0    -0.2     1.5     3.8
## x[1,24]          -0.1     0.2  1.3    -2.2    -1.0    -0.1     0.7     2.2
## x[1,25]           0.5     0.2  1.3    -1.9    -0.3     0.4     1.0     3.2
## x[1,26]           0.6     0.1  1.2    -1.5    -0.1     0.6     1.1     2.8
## x[1,27]           0.3     0.1  1.0    -2.3    -0.3     0.3     1.0     2.1
## x[1,28]           0.2     0.1  1.0    -2.0    -0.1     0.3     0.8     2.0
## x[1,29]           0.0     0.2  1.2    -2.0    -0.5     0.1     0.9     2.1
## x[1,30]           0.0     0.1  1.0    -1.7    -0.6     0.0     0.4     2.4
## x[1,31]          -0.3     0.1  1.0    -2.0    -0.8    -0.3     0.3     1.7
## x[1,32]          -0.5     0.1  1.2    -3.3    -1.2    -0.7     0.3     1.6
## x[1,33]          -0.7     0.2  1.2    -3.0    -1.3    -0.7     0.1     1.3
## x[1,34]          -0.4     0.1  1.3    -2.3    -1.2    -0.5     0.2     2.3
## x[1,35]          -0.6     0.2  1.4    -2.8    -1.3    -0.4     0.1     1.4
## x[1,36]          -0.5     0.2  1.5    -3.4    -1.1    -0.2     0.5     1.8
## x[1,37]           0.1     0.2  1.3    -2.3    -0.7     0.2     0.8     2.5
## x[1,38]           0.0     0.2  1.6    -3.7    -0.8    -0.1     1.2     2.2
## x[1,39]           0.3     0.2  1.1    -1.7    -0.1     0.3     0.8     2.4
## x[1,40]           0.3     0.2  1.1    -1.9    -0.4     0.4     1.0     2.8
## x[1,41]           0.2     0.1  1.1    -2.2    -0.3     0.3     1.1     2.0
## x[1,42]          -0.1     0.1  1.2    -2.2    -1.1     0.0     0.7     1.9
## x[1,43]          -0.4     0.2  1.3    -2.2    -1.3    -0.2     0.1     2.9
## x[1,44]          -0.4     0.2  1.3    -3.0    -0.8    -0.3     0.2     2.3
## x[1,45]          -0.1     0.2  1.3    -2.5    -1.2    -0.1     0.7     2.5
## x[1,46]          -0.3     0.2  1.3    -3.3    -0.9    -0.2     0.4     2.0
## x[1,47]          -0.5     0.1  1.5    -3.1    -1.5    -0.4     0.4     2.5
## x[1,48]          -0.3     0.2  1.3    -2.9    -0.9    -0.5     0.6     2.3
## x[1,49]           0.2     0.3  1.5    -2.7    -0.7     0.1     1.1     3.0
## x[1,50]          -0.1     0.2  1.3    -2.3    -1.1    -0.1     0.8     1.8
## x[1,51]          -0.1     0.3  1.3    -2.4    -1.0    -0.3     0.8     2.3
## x[1,52]          -0.5     0.1  1.2    -3.5    -1.0    -0.3     0.0     1.1
## x[1,53]          -0.4     0.2  1.2    -2.4    -1.5    -0.1     0.3     1.4
## x[1,54]          -0.5     0.1  0.8    -2.1    -0.9    -0.5     0.0     0.9
## x[1,55]          -0.5     0.2  1.1    -2.4    -1.3    -0.3     0.3     1.1
## x[1,56]          -0.2     0.2  1.3    -2.5    -0.8    -0.2     0.7     1.6
## x[1,57]           0.3     0.2  1.3    -1.9    -0.6     0.2     1.3     2.0
## x[1,58]           0.1     0.2  1.2    -2.3    -0.5     0.1     1.0     1.9
## x[1,59]           0.3     0.3  1.5    -2.2    -0.5     0.5     1.4     2.8
## x[1,60]           0.2     0.3  1.2    -2.5    -0.5     0.3     1.0     2.4
## x[1,61]           0.1     0.3  1.1    -2.0    -0.5     0.0     0.5     2.4
## x[1,62]           0.4     0.2  1.2    -1.7    -0.3     0.3     1.1     2.4
## x[1,63]           0.4     NaN  1.5    -2.3    -0.5     0.3     1.2     3.0
## x[1,64]           0.6     0.1  1.0    -0.8     0.1     0.7     1.1     3.0
## x[1,65]           0.6     0.1  1.4    -1.7    -0.2     0.5     1.5     3.2
## x[1,66]           0.4     0.3  1.3    -1.6    -0.5     0.4     1.1     2.4
## x[1,67]           0.5     0.1  1.2    -1.5    -0.4     0.2     1.4     3.4
## x[1,68]           0.2     0.2  1.2    -1.6    -0.7     0.1     1.0     2.9
## x[1,69]           0.0     0.2  1.2    -2.4    -0.8    -0.1     0.6     2.2
## x[1,70]          -0.4     0.1  0.9    -2.5    -1.0    -0.4     0.2     1.1
## x[1,71]          -1.2     0.1  1.2    -3.3    -2.0    -1.0    -0.5     0.8
## x[1,72]          -1.7     0.2  1.5    -4.8    -2.4    -1.6    -0.6     0.1
## x[1,73]          -2.0     0.2  1.6    -5.7    -2.6    -1.7    -0.9     0.0
## x[1,74]          -2.5     0.2  1.9    -7.3    -3.5    -2.1    -1.3    -0.1
## x[1,75]          -2.9     0.2  2.1    -6.2    -4.3    -2.4    -1.3    -0.1
## x[1,76]          -3.1     0.2  1.6    -6.7    -4.2    -3.2    -1.6    -0.5
## x[1,77]          -2.9     0.2  1.3    -5.6    -3.7    -2.8    -2.0    -0.8
## x[1,78]          -2.5     0.2  1.4    -4.8    -3.4    -2.6    -1.8     0.3
## x[1,79]          -1.6     0.2  1.1    -3.6    -2.3    -1.6    -0.8     0.0
## x[1,80]          -0.2     0.2  1.0    -1.6    -0.9    -0.2     0.4     1.8
## x[1,81]           2.6     0.2  1.6     0.5     1.6     2.3     3.5     6.0
## x[1,82]           3.5     0.2  1.7     0.7     2.4     3.5     4.8     7.0
## x[1,83]           3.8     0.2  1.9     1.3     2.2     3.2     5.5     7.1
## x[1,84]           3.5     0.2  1.8     0.9     2.1     3.3     4.7     6.8
## x[1,85]           1.6     0.2  1.1    -0.1     0.9     1.6     2.3     3.9
## x[1,86]           1.0     0.3  1.5    -1.1     0.1     0.7     1.6     4.7
## x[1,87]           0.8     0.2  1.5    -1.6     0.1     0.5     1.5     4.0
## x[1,88]           0.4     0.2  1.3    -1.4    -0.2     0.3     1.0     3.5
## x[1,89]           0.2     0.2  1.3    -2.1    -0.4     0.1     0.6     2.7
## x[1,90]           0.1     0.2  1.4    -3.3    -0.5     0.2     0.9     2.2
## x[1,91]          -0.2     0.2  1.2    -2.9    -0.8     0.1     0.7     1.7
## x[1,92]          -0.1     0.1  1.3    -2.3    -0.9    -0.1     0.7     2.1
## x[1,93]          -0.2     0.1  1.0    -2.0    -0.9    -0.1     0.6     1.3
## x[1,94]           0.1     0.1  1.2    -2.2    -0.8     0.3     1.1     2.0
## x[1,95]           0.1     0.1  1.2    -1.6    -0.9    -0.2     1.1     2.2
## x[1,96]          -0.6     0.1  1.1    -2.5    -1.3    -0.6     0.2     1.4
## x[1,97]          -1.0     0.1  1.2    -3.1    -1.7    -1.2    -0.3     0.9
## x[1,98]          -0.8     0.1  1.0    -2.5    -1.7    -0.5    -0.1     0.5
## x[1,99]          -0.7     0.1  1.1    -2.7    -1.3    -0.6     0.1     1.3
## x[1,100]         -0.5     0.2  1.3    -2.5    -1.3    -0.6     0.0     2.0
## x[1,101]         -0.4     0.3  1.2    -2.2    -1.1    -0.7     0.2     2.1
## x[1,102]         -0.4     0.1  1.0    -2.1    -1.2    -0.4     0.3     1.2
## x[1,103]         -0.3     0.1  1.1    -2.2    -0.9    -0.4     0.4     1.4
## x[1,104]          0.0     0.1  1.1    -2.1    -0.8     0.1     0.7     2.2
## x[1,105]          0.9     0.2  1.7    -1.4    -0.4     0.6     1.7     5.0
## x[1,106]          1.2     0.2  1.5    -0.9     0.2     0.6     2.2     4.6
## x[1,107]          1.2     0.3  1.6    -0.8     0.1     1.0     2.0     5.6
## x[1,108]          0.8     0.2  1.0    -1.3     0.4     0.9     1.5     2.6
## x[1,109]          0.3     0.2  1.2    -1.7    -0.5     0.2     0.8     2.7
## x[1,110]         -0.1     0.1  1.2    -1.7    -1.0    -0.2     0.5     2.3
## x[1,111]         -0.6     0.2  1.1    -2.6    -1.3    -0.5     0.2     1.1
## x[1,112]         -1.0     0.1  1.0    -3.3    -1.7    -0.9    -0.3     0.4
## x[1,113]         -0.8     0.2  1.2    -2.7    -1.6    -0.8    -0.1     0.9
## x[1,114]         -0.4     0.2  1.1    -2.8    -1.1    -0.2     0.3     1.4
## x[1,115]         -0.3     0.1  1.2    -2.3    -1.0    -0.2     0.6     1.6
## x[1,116]         -0.1     0.2  1.1    -2.4    -0.5    -0.1     0.4     2.0
## x[1,117]          0.2     0.2  1.4    -1.6    -0.9     0.1     0.9     2.6
## x[1,118]          0.6     0.2  1.3    -2.0    -0.2     0.5     1.2     3.4
## x[1,119]          0.7     0.2  1.3    -1.1    -0.3     0.6     1.4     2.8
## x[1,120]          0.5     0.1  0.9    -1.0    -0.2     0.4     1.2     2.1
## x[1,121]          0.2     0.2  1.0    -1.9    -0.4     0.3     0.9     1.6
## x[1,122]          0.2     0.1  1.0    -1.6    -0.4     0.2     0.8     1.8
## x[1,123]          0.1     0.2  1.3    -2.2    -0.6     0.5     1.0     1.7
## x[1,124]          0.4     0.2  1.2    -1.7    -0.3     0.4     1.3     2.6
## x[1,125]          0.3     0.2  1.2    -1.9    -0.6     0.1     0.9     3.1
## x[1,126]         -0.5     0.1  0.9    -2.0    -0.9    -0.5    -0.1     1.6
## x[1,127]         -0.9     0.2  1.2    -3.0    -1.6    -1.0     0.0     1.4
## x[1,128]         -0.9     NaN  1.2    -2.9    -1.7    -0.8     0.0     1.3
## x[1,129]         -0.9     0.1  0.9    -2.6    -1.5    -0.9    -0.3     0.7
## x[1,130]         -0.5     0.2  1.3    -2.9    -1.3    -0.4     0.2     1.7
## x[1,131]         -0.4     0.1  0.9    -2.0    -0.9    -0.4     0.1     1.1
## x[1,132]         -0.4     0.1  1.4    -2.8    -1.1    -0.4     0.2     2.9
## x[1,133]         -0.9     0.1  1.4    -3.3    -1.6    -0.9    -0.2     1.4
## x[1,134]         -1.1     0.2  1.1    -3.0    -1.9    -1.1    -0.3     0.6
## x[1,135]         -1.4     0.2  1.1    -3.8    -2.3    -1.1    -0.5     0.2
## x[1,136]         -1.6     0.2  1.5    -4.2    -2.6    -1.6    -0.8     0.5
## x[1,137]         -1.1     0.2  1.1    -3.0    -1.9    -1.2    -0.1     0.6
## x[1,138]         -1.2     0.1  0.9    -2.9    -1.7    -1.0    -0.4     0.2
## x[1,139]         -0.3     0.2  1.4    -3.1    -1.1    -0.2     0.2     2.5
## x[1,140]          0.3     0.2  1.0    -1.2    -0.3     0.1     0.6     2.2
## x[1,141]          1.4     0.2  1.3    -0.5     0.5     1.2     2.1     4.0
## x[1,142]          1.2     0.2  1.5    -1.4     0.2     1.1     2.1     3.7
## x[1,143]          0.3     0.2  1.6    -2.9    -0.8     0.4     0.9     3.1
## x[1,144]         -0.9     0.2  1.4    -3.0    -1.6    -1.0    -0.3     2.2
## x[1,145]         -2.9     0.3  1.7    -6.8    -3.8    -2.8    -1.6    -0.6
## x[1,146]         -3.1     0.2  1.7    -6.6    -4.1    -2.6    -1.8    -0.8
## x[1,147]         -1.2     0.2  1.1    -3.2    -2.0    -1.1    -0.5     0.6
## x[1,148]         -0.2     0.2  1.2    -2.2    -1.1    -0.3     0.6     1.9
## x[1,149]          0.1     0.1  1.1    -1.6    -0.4     0.1     0.9     2.2
## x[1,150]          0.8     0.1  1.2    -1.1    -0.1     0.7     1.8     3.0
## x[1,151]          2.1     0.2  1.5    -0.1     1.0     2.0     3.1     4.7
## x[1,152]          3.6     0.3  2.0     1.0     2.1     3.5     4.7     8.1
## x[1,153]          3.5     0.3  1.8     1.2     2.1     3.2     4.4     7.2
## x[1,154]          1.9     0.3  1.4     0.2     0.8     1.6     2.6     4.8
## x[1,155]          0.8     0.1  1.0    -0.5     0.1     0.6     1.5     2.2
## x[1,156]          0.4     0.1  1.3    -1.4    -0.5     0.3     1.1     2.8
## x[1,157]          0.7     0.1  1.4    -1.5    -0.5     0.6     1.8     3.5
## x[1,158]          0.5     0.3  1.5    -2.8    -0.3     0.6     1.5     2.8
## x[1,159]          0.4     0.2  1.3    -1.8    -0.4     0.5     1.2     2.9
## x[1,160]          0.1     0.2  1.3    -2.6    -0.4     0.3     0.9     1.8
## x[1,161]         -0.2     0.1  0.9    -2.4    -0.6     0.1     0.4     1.2
## x[1,162]         -0.1     0.1  1.1    -2.0    -0.8     0.0     0.6     2.1
## x[1,163]         -0.4     0.1  1.0    -2.1    -1.1    -0.3     0.1     1.4
## x[1,164]         -0.4     0.1  1.4    -3.0    -1.0    -0.2     0.3     2.4
## x[1,165]         -0.2     0.2  1.1    -2.0    -0.8    -0.2     0.4     1.8
## x[1,166]         -0.2     0.1  1.5    -2.4    -1.1    -0.4     0.4     3.0
## x[1,167]         -0.2     0.1  1.1    -1.7    -0.8    -0.4     0.4     2.4
## x[1,168]         -0.5     0.1  1.2    -2.5    -1.4    -0.6     0.4     1.5
## x[1,169]         -0.8     0.1  1.5    -3.2    -2.1    -0.7     0.1     1.9
## x[1,170]         -1.0     0.2  1.5    -3.2    -2.1    -1.3    -0.2     1.8
## x[1,171]         -1.2     0.1  1.4    -3.6    -2.4    -1.0     0.0     0.6
## x[1,172]         -1.1     0.2  1.6    -4.5    -1.9    -0.9     0.2     1.3
## x[1,173]         -0.3     0.1  1.1    -2.5    -0.6    -0.2     0.3     1.1
## x[1,174]          0.4     0.2  1.4    -2.0    -0.6     0.5     1.2     3.4
## x[1,175]          0.7     0.2  1.6    -2.5    -0.3     0.5     1.5     3.8
## x[1,176]          0.6     0.1  1.1    -1.1     0.0     0.6     1.2     3.0
## x[1,177]          1.0     0.1  1.1    -1.0     0.4     1.1     1.8     2.9
## x[1,178]          0.9     0.1  1.2    -0.9     0.3     0.7     1.3     3.3
## x[1,179]          0.3     0.1  1.1    -2.1    -0.2     0.4     0.9     2.2
## x[1,180]         -0.5     0.2  1.3    -2.9    -1.3    -0.4     0.4     1.4
## x[1,181]         -1.9     0.2  1.5    -5.1    -2.7    -1.4    -0.8     0.0
## x[1,182]         -2.4     0.2  1.2    -4.6    -3.2    -2.4    -1.5    -0.2
## x[1,183]         -2.5     0.2  1.2    -4.7    -3.0    -2.4    -1.6    -0.5
## x[1,184]         -2.8     0.2  1.6    -6.3    -3.6    -2.5    -1.7    -0.5
## x[1,185]         -2.6     0.3  1.8    -6.9    -3.4    -2.5    -1.2    -0.2
## x[1,186]         -2.5     0.2  1.6    -6.5    -3.0    -2.5    -1.4    -0.2
## x[1,187]         -1.7     0.2  1.6    -5.4    -2.3    -1.3    -0.8     0.6
## x[1,188]          0.0     0.1  1.5    -2.9    -1.0     0.0     0.6     2.5
## x[1,189]          1.5     0.1  1.4    -0.6     0.7     1.3     1.9     4.2
## x[1,190]          4.0     0.3  2.5     1.1     2.3     3.5     5.1    11.7
## x[1,191]          4.7     0.3  2.2     1.4     3.2     4.5     5.7     8.8
## x[1,192]          3.9     0.2  2.2     1.1     2.2     3.7     5.5     7.7
## x[1,193]          1.1     0.2  1.6    -1.9     0.1     0.7     2.1     4.5
## x[1,194]         -0.1     0.2  1.4    -3.2    -1.0     0.2     0.8     2.1
## x[1,195]         -0.6     0.2  1.5    -3.5    -1.5    -0.8     0.3     2.1
## x[1,196]         -1.1     0.2  1.4    -3.8    -1.9    -1.2    -0.6     1.8
## x[1,197]         -1.4     0.1  1.4    -4.2    -2.5    -1.4    -0.4     0.9
## x[1,198]         -1.5     0.1  1.4    -3.9    -2.4    -1.4    -0.5     0.9
## x[1,199]         -1.5     0.1  1.4    -4.3    -2.5    -1.4    -0.4     1.1
## x[1,200]         -1.0     0.2  1.4    -4.2    -2.0    -0.6     0.0     1.1
## x[1,201]          0.7     0.1  1.2    -2.2     0.0     0.9     1.3     2.5
## x[1,202]          3.3     0.3  2.0     0.9     1.9     3.0     3.8     8.3
## x[1,203]          5.6     0.4  2.7     1.7     3.3     5.8     7.4    10.8
## x[1,204]          6.8     0.3  3.3     1.7     4.2     6.3     9.3    14.1
## x[1,205]          7.2     0.3  3.5     1.9     4.3     7.0     9.5    13.6
## x[1,206]          5.4     0.3  3.0     1.7     2.9     5.0     7.3    11.1
## x[1,207]          4.7     0.2  2.7     1.4     2.5     4.1     7.0    10.1
## x[1,208]          4.9     0.2  2.6     1.7     2.7     4.4     6.5    10.7
## x[1,209]          2.0     0.2  1.6    -0.3     1.0     1.9     3.1     5.4
## x[1,210]         -0.1     0.1  1.2    -3.1    -0.8     0.1     0.6     1.4
## x[1,211]         -1.7     0.2  1.5    -4.8    -2.3    -1.3    -0.9     0.4
## x[1,212]         -3.0     0.3  2.1    -6.9    -4.4    -2.7    -1.3    -0.5
## x[1,213]         -3.7     0.2  2.0    -7.0    -5.0    -3.4    -2.2    -1.3
## x[1,214]         -3.7     0.3  1.9    -7.4    -5.1    -3.7    -2.5    -0.3
## x[1,215]         -3.6     0.2  1.8    -7.3    -4.6    -3.5    -2.3    -0.6
## x[1,216]         -2.3     0.2  1.9    -6.4    -3.3    -1.8    -1.0     0.2
## x[1,217]          0.4     0.2  1.2    -1.6    -0.2     0.3     0.8     3.4
## x[1,218]          1.3     0.2  1.3    -0.9     0.8     1.4     1.8     4.6
## x[1,219]          1.0     0.1  1.2    -0.9     0.1     1.0     1.9     3.1
## x[1,220]          0.4     0.2  1.3    -2.5    -0.1     0.4     1.0     2.8
## x[1,221]          0.0     0.2  1.2    -2.3    -0.6     0.1     0.6     2.0
## x[1,222]         -0.8     0.1  1.3    -3.6    -1.7    -0.6     0.3     0.9
## x[1,223]         -1.4     0.2  1.4    -4.7    -2.0    -1.0    -0.6     1.0
## x[1,224]         -1.5     0.0  1.5    -4.5    -2.3    -1.1    -0.5     0.8
## x[1,225]         -1.2     0.0  1.2    -3.4    -2.1    -1.0    -0.5     0.7
## x[1,226]         -1.7     0.1  1.4    -4.8    -2.5    -1.3    -0.6     0.3
## x[1,227]         -1.3     0.2  1.4    -4.4    -1.9    -1.2    -0.4     0.7
## x[1,228]         -1.4     0.1  1.9    -5.5    -2.3    -1.1    -0.1     1.4
## x[1,229]         -0.3     0.2  1.4    -3.3    -1.0    -0.1     0.5     2.0
## x[1,230]          0.1     0.2  1.2    -2.0    -0.4     0.0     0.6     2.2
## x[1,231]          0.1     0.1  1.4    -2.9    -0.8     0.1     1.2     2.2
## x[1,232]          0.2     0.1  1.3    -2.3    -0.6     0.4     1.1     2.1
## x[1,233]          0.4     0.2  1.3    -2.0    -0.4     0.6     1.3     2.5
## x[1,234]          0.2     0.1  1.2    -1.8    -0.4     0.2     0.7     2.8
## x[1,235]          0.3     0.2  1.5    -2.0    -0.5     0.4     1.4     2.9
## x[1,236]         -0.1     0.1  1.1    -2.1    -0.9     0.0     0.7     1.8
## x[1,237]         -0.3     0.2  1.2    -3.0    -0.8    -0.2     0.3     1.7
## x[1,238]         -0.4     0.2  1.4    -3.2    -1.2    -0.5     0.7     2.3
## x[1,239]         -0.5     0.2  1.2    -3.2    -1.0    -0.5     0.0     1.7
## x[1,240]         -0.4     0.2  1.5    -2.8    -1.1    -0.4     0.1     2.3
## x[1,241]         -0.2     0.2  1.4    -2.5    -1.2     0.0     0.4     2.2
## x[1,242]          0.2     0.3  1.5    -2.6    -0.6     0.1     1.0     3.2
## x[1,243]          0.1     0.2  1.3    -2.0    -0.6     0.0     0.6     3.1
## x[1,244]          0.0     0.2  1.3    -2.4    -0.7    -0.1     0.9     1.8
## x[1,245]         -0.1     0.3  1.7    -3.3    -0.9    -0.3     0.9     2.3
## x[1,246]          0.0     0.2  1.6    -2.6    -0.9    -0.2     0.8     2.4
## x[1,247]         -0.5     0.2  1.7    -4.1    -1.6    -0.3     0.5     2.1
## x[1,248]         -0.2     0.2  1.6    -3.0    -1.0    -0.4     0.1     3.6
## x[1,249]         -0.1     0.2  1.4    -2.4    -0.8    -0.3     0.6     3.6
## x[1,250]         -0.3     0.2  1.1    -2.1    -1.0    -0.3     0.4     1.8
## x[1,251]         -0.6     0.2  1.2    -2.5    -1.2    -0.6     0.0     1.5
## x[1,252]         -0.7     0.3  1.4    -4.2    -1.0    -0.3     0.3     0.8
## x[1,253]         -0.5     0.2  1.3    -3.5    -0.9    -0.3     0.2     1.5
## x[1,254]         -0.3     0.2  1.1    -3.0    -0.9    -0.3     0.5     1.4
## x[1,255]          0.0     0.2  1.4    -2.9    -0.9     0.2     0.9     1.7
## x[1,256]         -0.2     0.1  1.3    -2.8    -1.2    -0.3     0.7     2.2
## x[1,257]         -0.2     0.2  1.1    -2.4    -0.7    -0.2     0.4     1.6
## x[1,258]         -0.2     0.1  1.0    -1.9    -0.9    -0.1     0.5     1.7
## x[1,259]          0.0     0.2  1.5    -3.2    -1.0     0.1     1.1     2.8
## x[1,260]         -0.1     0.2  1.4    -2.6    -1.0    -0.3     0.8     2.5
## x[1,261]         -0.2     0.1  1.0    -2.1    -1.1    -0.2     0.3     1.8
## x[1,262]         -0.5     0.1  1.1    -2.3    -1.3    -0.6     0.2     1.7
## x[1,263]         -0.9     0.2  1.1    -2.6    -2.0    -0.6    -0.2     1.4
## x[1,264]         -1.1     0.1  1.1    -2.8    -1.7    -1.3    -0.3     0.8
## x[1,265]         -1.9     0.2  1.4    -4.9    -2.5    -1.5    -1.1     0.3
## x[1,266]         -1.9     0.2  1.9    -7.0    -2.7    -1.6    -0.8     0.9
## x[1,267]         -1.4     0.0  1.2    -3.4    -2.2    -1.5    -0.6     1.0
## x[1,268]         -1.3     0.1  1.1    -3.5    -1.9    -1.2    -0.6     0.7
## x[1,269]         -0.8     0.1  0.9    -2.2    -1.5    -0.8    -0.3     0.4
## x[1,270]         -0.6     0.2  1.4    -3.0    -1.2    -0.5     0.2     2.3
## x[1,271]         -0.4     0.2  1.2    -3.1    -1.1    -0.1     0.1     1.5
## x[1,272]          0.3     0.1  1.1    -1.9    -0.4     0.1     1.2     2.3
## x[1,273]          1.2     0.2  1.3    -0.5     0.3     1.0     1.9     3.5
## x[1,274]          1.9     0.2  1.3     0.2     0.9     1.9     2.5     5.2
## x[1,275]          2.5     0.2  1.5     0.1     1.3     2.3     3.8     5.3
## x[1,276]          2.7     0.2  1.5     0.4     1.4     2.6     3.7     5.6
## x[1,277]          1.8     0.2  1.3    -0.3     0.7     2.0     2.6     4.4
## x[1,278]          1.1     0.2  1.3    -0.6     0.1     0.8     1.9     4.3
## x[1,279]          0.7     0.1  1.4    -1.6    -0.1     0.4     1.5     3.9
## x[1,280]          0.7     0.2  1.2    -1.1    -0.1     0.7     1.4     3.0
## x[1,281]          0.7     0.2  1.6    -1.7    -0.4     0.3     1.8     4.5
## x[1,282]          0.1     0.2  1.5    -3.2    -0.4     0.1     0.7     2.4
## x[1,283]         -0.3     0.2  1.3    -2.0    -1.2    -0.3     0.2     1.3
## x[1,284]         -0.5     0.2  1.2    -2.7    -1.3    -0.6     0.2     1.7
## x[1,285]         -1.1     0.2  1.2    -3.3    -1.8    -1.0    -0.4     1.0
## x[1,286]         -1.3     0.1  1.4    -3.8    -2.3    -1.2    -0.4     1.1
## x[1,287]         -1.3     0.3  1.5    -4.4    -1.8    -1.2    -0.4     1.0
## x[1,288]         -1.0     0.2  1.1    -2.9    -1.5    -1.0    -0.2     1.0
## x[1,289]         -1.4     0.2  1.2    -3.1    -2.2    -1.7    -0.7     1.1
## x[1,290]         -1.2     0.1  1.4    -4.2    -2.0    -1.1    -0.1     0.9
## x[1,291]         -0.7     0.1  1.2    -2.8    -1.8    -0.6     0.0     1.9
## x[1,292]          0.2     0.1  1.1    -1.5    -0.4     0.0     0.8     2.4
## x[1,293]          0.5     0.1  1.6    -2.1    -0.6     0.4     1.6     3.9
## x[1,294]          0.6     0.3  1.5    -1.7    -0.6     0.3     1.4     3.5
## x[1,295]          0.7     0.2  1.4    -1.6    -0.3     0.7     1.4     3.5
## x[1,296]          0.4     0.2  1.3    -2.4    -0.3     0.4     1.1     2.8
## x[1,297]          0.2     0.2  1.5    -1.8    -0.8     0.2     1.2     3.8
## x[1,298]          0.6     0.2  1.2    -1.3    -0.2     0.4     1.2     3.0
## x[1,299]          1.1     0.2  1.6    -1.9     0.2     0.9     2.3     4.4
## x[1,300]          1.9     0.1  1.3    -0.2     1.1     1.9     2.4     4.6
## x[1,301]          2.4     0.2  1.8    -0.4     1.0     2.5     3.5     5.5
## x[1,302]          2.8     0.2  1.8     0.2     1.4     2.4     4.0     6.1
## x[1,303]          2.9     0.2  1.7     0.3     1.6     3.0     3.7     5.7
## x[1,304]          1.3     0.2  1.4    -0.8     0.3     1.1     2.2     4.1
## x[1,305]         -0.3     0.2  1.7    -3.7    -1.2     0.1     0.7     2.6
## x[1,306]         -0.9     0.2  1.6    -4.4    -1.4    -0.6     0.1     1.2
## x[1,307]         -1.2     0.2  2.0    -5.8    -2.2    -1.0    -0.1     2.1
## x[1,308]         -1.2     NaN  1.8    -4.8    -2.6    -1.2     0.0     2.4
## x[1,309]         -1.5     0.1  1.5    -4.7    -2.6    -1.1    -0.5     0.8
## x[1,310]         -1.0     0.1  1.3    -3.4    -1.7    -1.0     0.0     1.6
## x[1,311]         -0.2     0.1  1.0    -2.0    -1.0    -0.3     0.5     1.4
## x[1,312]          0.2     0.1  1.2    -2.2    -0.5     0.0     0.8     2.5
## x[1,313]          0.5     0.2  1.2    -1.6    -0.2     0.5     1.1     2.4
## x[1,314]          0.7     0.2  1.1    -1.0    -0.1     0.6     1.3     2.7
## x[1,315]          0.9     0.2  1.4    -1.6    -0.1     0.7     1.8     3.3
## x[1,316]          0.8     0.1  1.3    -1.0     0.2     0.6     1.1     3.9
## x[1,317]          0.5     0.1  1.6    -1.5    -0.6    -0.1     1.3     4.4
## x[1,318]          0.1     0.2  1.4    -2.4    -1.0     0.0     0.9     2.9
## x[1,319]          0.0     0.2  1.1    -1.9    -0.8     0.1     0.8     1.6
## x[1,320]         -0.1     0.2  1.0    -2.2    -0.4    -0.1     0.6     1.2
## x[1,321]         -0.2     0.1  1.2    -3.2    -0.9     0.1     0.6     1.5
## x[1,322]         -0.9     0.2  1.5    -4.1    -1.6    -0.7    -0.2     1.9
## x[1,323]         -1.2     0.1  1.3    -3.8    -1.8    -1.1    -0.5     1.5
## x[1,324]         -0.9     0.2  1.7    -4.3    -1.9    -0.9     0.0     3.2
## x[1,325]         -0.7     0.1  1.2    -2.5    -1.5    -1.1    -0.4     2.2
## x[1,326]         -0.6     0.1  1.3    -2.5    -1.3    -0.7     0.4     2.1
## x[1,327]         -0.5     0.2  1.2    -2.9    -1.4    -0.2     0.3     1.7
## x[1,328]         -0.3     0.2  1.2    -2.5    -1.2    -0.1     0.7     1.4
## x[1,329]         -0.3     0.2  1.2    -2.5    -1.0    -0.4     0.3     2.2
## x[1,330]         -0.4     0.1  1.4    -3.0    -1.1    -0.3     0.3     2.4
## x[1,331]         -0.5     0.1  1.6    -3.1    -1.8    -0.5     0.7     2.7
## x[1,332]         -0.1     0.2  1.4    -2.0    -1.0    -0.5     0.9     2.9
## x[1,333]         -0.5     0.2  1.0    -2.8    -1.2    -0.3     0.2     0.8
## x[1,334]         -0.4     0.2  1.2    -2.7    -0.9    -0.5     0.4     1.4
## x[1,335]         -0.3     0.2  1.2    -3.4    -0.8    -0.2     0.5     1.9
## x[1,336]         -0.9     0.2  1.3    -4.1    -1.4    -0.9    -0.1     1.1
## x[1,337]         -2.5     0.2  1.7    -6.1    -3.7    -1.8    -1.2    -0.3
## x[1,338]         -3.0     0.2  1.8    -6.3    -4.2    -2.5    -1.7    -0.5
## x[1,339]         -3.4     0.2  2.1    -7.6    -4.4    -3.3    -1.9    -0.5
## x[1,340]         -2.8     0.1  1.4    -5.4    -3.7    -2.6    -1.7    -0.6
## x[1,341]         -2.0     0.1  1.4    -4.2    -3.2    -1.8    -0.8     0.4
## x[1,342]         -1.2     0.2  1.4    -4.2    -1.8    -1.1    -0.3     0.7
## x[1,343]          0.4     0.1  1.2    -1.9    -0.2     0.3     1.1     2.8
## x[1,344]          2.1     0.2  1.5    -0.2     0.9     2.0     3.3     4.8
## x[1,345]          3.5     0.2  2.0     0.6     1.6     3.4     4.6     7.0
## x[1,346]          3.7     0.1  1.4     1.2     2.7     3.5     4.7     6.3
## x[1,347]          3.9     0.2  2.0     0.8     2.4     4.2     5.1     7.6
## x[1,348]          3.6     0.2  2.0     0.8     2.2     3.5     4.7     7.8
## x[1,349]          3.0     0.3  1.8     0.4     2.0     2.8     4.1     6.4
## x[1,350]          2.2     0.1  1.7    -1.0     1.0     2.0     3.2     5.2
## x[1,351]          1.5     0.1  1.3    -1.2     0.7     1.5     2.2     3.8
## x[1,352]          0.9     0.1  1.6    -2.2    -0.2     1.1     1.9     4.5
## x[1,353]          0.2     0.0  1.4    -2.3    -0.7     0.4     0.8     3.5
## x[1,354]         -0.4     0.2  1.2    -2.7    -1.1    -0.5     0.7     1.5
## x[1,355]         -0.4     0.1  1.3    -2.4    -1.4    -0.6     0.3     1.9
## x[1,356]         -0.3     0.2  1.0    -1.8    -1.2    -0.5     0.1     1.6
## x[1,357]         -0.7     0.1  1.3    -2.9    -1.5    -0.7     0.3     1.4
## x[1,358]         -0.9     0.2  1.3    -2.9    -2.1    -0.9     0.1     1.1
## x[1,359]         -1.2     0.2  1.2    -3.4    -1.8    -0.9    -0.2     0.4
## x[1,360]         -1.5     0.2  1.1    -3.2    -2.4    -1.3    -0.6     0.3
## x[1,361]         -1.4     0.1  0.9    -3.0    -2.1    -1.2    -0.8    -0.2
## x[1,362]         -1.7     0.1  1.2    -3.7    -2.7    -1.6    -0.6     0.2
## x[1,363]         -1.7     NaN  1.3    -4.0    -2.6    -1.5    -0.8     0.4
## x[1,364]         -1.1     0.2  1.5    -3.3    -2.2    -1.1    -0.2     1.5
## x[1,365]         -0.6     0.2  1.7    -3.6    -1.8    -0.4     0.4     2.8
## x[1,366]         -0.2     0.2  1.4    -2.1    -1.3    -0.3     0.7     2.5
## x[1,367]         -0.2     0.1  1.3    -2.5    -1.4     0.0     0.5     2.0
## x[1,368]         -0.2     NaN  1.2    -2.7    -1.0     0.0     0.7     1.5
## x[1,369]          0.3     0.1  0.9    -1.4     0.0     0.4     1.0     1.7
## x[1,370]          1.1     0.3  1.2    -1.8     0.5     1.1     1.8     3.4
## x[1,371]          1.7     0.1  1.3    -0.1     0.8     1.5     2.6     3.9
## x[1,372]          1.3     0.1  1.4    -1.5     0.6     1.3     1.9     3.7
## x[1,373]          0.7     0.1  1.1    -2.0     0.0     0.6     1.1     2.6
## x[1,374]          0.1     0.2  1.5    -2.3    -0.7     0.2     0.8     2.5
## x[1,375]          0.2     0.1  1.3    -2.1    -0.5     0.0     1.0     3.2
## x[1,376]         -0.5     0.2  1.5    -3.5    -1.0    -0.5     0.2     1.5
## x[1,377]         -1.1     0.3  1.6    -4.5    -1.5    -0.8    -0.3     0.7
## x[1,378]         -1.2     0.2  1.6    -4.6    -1.8    -0.9    -0.3     0.6
## x[1,379]         -0.9     0.1  1.2    -3.6    -1.3    -0.6    -0.2     0.7
## x[1,380]         -0.5     0.1  1.2    -2.9    -1.3    -0.4     0.4     1.4
## x[1,381]          0.4     0.2  1.2    -1.5    -0.3     0.2     1.0     2.9
## x[1,382]          1.1     0.2  1.4    -1.0    -0.1     1.0     2.2     3.8
## x[1,383]          1.2     0.2  1.2    -1.0     0.5     1.1     2.1     3.4
## x[1,384]          0.9     0.1  1.2    -0.8     0.0     0.7     1.5     3.5
## x[1,385]          0.7     0.1  1.3    -1.6    -0.3     0.7     1.5     3.1
## x[1,386]          0.7     0.2  1.1    -1.3     0.0     0.6     1.5     2.9
## x[1,387]          0.1     0.1  1.1    -1.9    -0.4     0.0     1.0     2.2
## x[1,388]          0.0     0.1  1.2    -2.0    -0.9     0.0     0.5     2.0
## x[1,389]         -0.3     0.2  1.4    -3.7    -0.8    -0.4     0.3     2.3
## x[1,390]         -0.5     0.1  1.2    -2.6    -1.3    -0.4     0.1     1.7
## x[1,391]         -0.3     0.1  1.5    -3.1    -1.2    -0.2     0.6     2.1
## x[1,392]          0.3     0.1  1.0    -1.1    -0.5     0.2     1.0     2.0
## x[1,393]          1.1     NaN  1.2    -0.8     0.0     1.1     1.9     3.6
## x[1,394]          1.1     0.2  1.1    -0.6     0.5     1.0     1.7     3.4
## x[1,395]          1.3     0.2  1.5    -0.9    -0.1     1.2     2.5     4.0
## x[1,396]          1.7     0.2  1.3    -0.2     0.6     1.5     2.6     4.5
## x[1,397]          3.2     0.3  1.8     0.8     2.2     2.8     4.2     7.1
## x[1,398]          3.9     0.3  2.1     0.8     2.3     3.7     5.7     8.4
## x[1,399]          3.2     0.3  2.2     0.2     1.3     2.8     4.8     7.8
## x[1,400]          2.1     0.3  1.4     0.1     0.9     2.2     2.9     4.5
## x[1,401]          0.3     0.3  1.0    -1.3    -0.4     0.4     0.9     2.3
## x[1,402]         -0.9     0.2  1.6    -3.5    -1.9    -0.9     0.2     2.2
## x[1,403]         -1.2     0.1  1.4    -3.7    -2.0    -0.8    -0.4     1.3
## x[1,404]         -1.3     0.1  1.2    -3.9    -1.9    -1.3    -0.5     0.9
## x[1,405]         -2.1     0.1  1.5    -4.9    -3.1    -1.9    -1.0     0.3
## x[1,406]         -2.7     0.2  1.5    -6.1    -3.7    -2.7    -1.5    -0.6
## x[1,407]         -3.3     0.3  1.8    -6.7    -3.8    -3.0    -1.9    -1.0
## x[1,408]         -2.6     0.2  1.7    -5.8    -3.9    -2.1    -1.2    -0.4
## x[1,409]         -1.5     0.2  1.4    -3.6    -2.9    -1.1    -0.5     0.8
## x[1,410]         -0.1     0.1  1.0    -1.8    -0.6    -0.3     0.3     2.4
## x[1,411]          0.9     0.2  1.4    -1.6     0.0     0.7     2.1     3.3
## x[1,412]          2.0     0.2  1.7    -0.4     0.9     1.8     2.9     5.9
## x[1,413]          2.4     0.1  1.3     0.6     1.4     2.0     3.3     5.4
## x[1,414]          2.1     0.1  1.4     0.2     1.3     2.0     2.8     5.1
## x[1,415]          1.2     0.2  1.4    -1.0     0.2     1.3     2.0     3.9
## x[1,416]          0.5     0.2  1.0    -1.7     0.1     0.4     1.1     2.2
## x[1,417]         -0.7     0.2  1.1    -2.4    -1.3    -0.8     0.1     1.8
## x[1,418]         -2.3     0.2  1.8    -4.9    -4.0    -2.0    -0.7     0.4
## x[1,419]         -3.1     0.2  1.9    -7.0    -4.3    -2.6    -1.6    -0.7
## x[1,420]         -2.5     0.1  1.4    -5.7    -3.3    -2.0    -1.6    -0.7
## x[1,421]         -0.7     0.2  1.2    -2.8    -1.4    -0.6     0.3     0.9
## x[1,422]          0.4     0.0  1.2    -2.1    -0.4     0.4     1.2     2.1
## x[1,423]          1.0     0.1  1.7    -1.8    -0.2     1.1     2.2     4.1
## x[1,424]          1.2     0.1  1.1    -0.4     0.6     1.2     2.0     3.2
## x[1,425]          1.3     0.2  1.1     0.1     0.5     1.2     1.7     3.6
## x[1,426]          1.0     0.1  1.0    -0.4     0.3     0.6     1.5     3.1
## x[1,427]          0.4     0.1  1.3    -1.2    -0.5     0.2     0.9     2.9
## x[1,428]         -0.2     0.2  1.7    -3.7    -1.5    -0.1     0.8     2.6
## x[1,429]         -0.8     0.1  1.2    -3.8    -1.3    -0.7    -0.1     1.9
## x[1,430]         -1.0     0.2  1.8    -3.1    -2.3    -1.2     0.0     3.2
## x[1,431]         -1.5     0.2  1.4    -3.6    -2.6    -1.5    -0.8     1.2
## x[1,432]         -1.8     0.2  1.5    -5.0    -2.6    -1.5    -1.1     1.3
## x[1,433]         -0.9     0.2  1.4    -3.5    -1.8    -0.9     0.0     1.6
## x[1,434]         -0.4     0.2  1.0    -1.9    -0.9    -0.6    -0.2     2.0
## x[1,435]          0.0     0.1  1.2    -2.1    -0.7    -0.1     0.8     2.3
## x[1,436]          0.0     0.1  0.9    -1.2    -0.5     0.1     0.5     1.9
## x[1,437]         -0.2     0.1  0.9    -2.1    -0.8    -0.1     0.3     1.3
## x[1,438]          0.1     0.1  1.3    -2.0    -0.7    -0.3     0.7     2.5
## x[1,439]          0.3     0.1  1.1    -1.8    -0.7     0.2     1.1     2.5
## x[1,440]          0.5     0.1  0.9    -0.9    -0.1     0.3     1.0     2.6
## x[1,441]          0.6     0.1  1.2    -1.6    -0.1     0.4     1.4     2.5
## x[1,442]          0.7     0.2  1.1    -0.6     0.0     0.4     1.3     3.3
## x[1,443]          1.0     0.1  1.6    -1.7     0.2     1.0     1.6     4.2
## x[1,444]          1.9     NaN  1.7    -0.4     1.0     1.6     2.9     5.5
## x[1,445]          3.1     0.1  1.7     0.5     1.7     3.0     4.4     6.1
## x[1,446]          3.6     0.2  2.3     0.8     1.8     3.0     4.9     8.0
## x[1,447]          3.3     0.2  1.7     1.1     2.0     3.2     4.3     7.1
## x[1,448]          2.6     0.2  1.7     0.7     1.6     2.3     3.1     6.0
## x[1,449]          1.5     0.2  1.4    -0.6     0.5     1.3     2.4     4.2
## x[1,450]          0.3     0.2  1.2    -1.3    -0.5     0.1     0.8     3.2
## x[1,451]         -0.7     0.1  1.2    -2.6    -1.6    -0.6     0.0     1.7
## x[1,452]         -1.6     0.1  1.4    -4.1    -2.2    -1.6    -1.0     0.8
## x[1,453]         -3.0     0.2  1.9    -7.2    -3.9    -2.7    -1.6    -0.3
## x[1,454]         -4.1     0.2  2.2    -8.4    -5.7    -4.0    -2.3    -0.6
## x[1,455]         -4.4     0.2  1.8    -7.8    -5.6    -4.5    -3.3    -1.6
## x[1,456]         -3.9     0.2  1.4    -6.8    -4.8    -4.0    -2.9    -1.3
## x[1,457]         -3.8     0.3  1.7    -6.5    -4.8    -4.2    -2.6    -0.2
## x[1,458]         -3.9     0.1  1.8    -8.3    -4.9    -3.4    -2.7    -1.8
## x[1,459]         -4.0     0.2  2.1    -9.3    -4.7    -3.8    -2.5    -1.2
## x[1,460]         -3.7     0.2  2.0    -8.7    -4.8    -3.1    -2.4    -1.3
## x[1,461]         -3.8     0.2  1.7    -7.9    -4.8    -3.4    -2.4    -1.5
## x[1,462]         -3.2     0.1  1.7    -6.9    -4.3    -3.2    -1.7    -1.0
## x[1,463]         -2.4     0.2  1.2    -5.3    -2.8    -2.3    -1.6    -0.8
## x[1,464]         -0.5     0.1  1.3    -2.5    -1.4    -0.5     0.3     2.0
## x[1,465]          1.6     0.1  1.1    -0.1     0.8     1.5     2.3     3.6
## x[1,466]          4.8     0.2  2.0     1.9     3.4     4.7     5.9     8.7
## x[1,467]          6.6     0.3  2.5     2.7     4.6     6.4     8.2    11.2
## x[1,468]          7.3     0.3  2.7     3.2     4.9     7.1     9.4    12.4
## x[1,469]          6.6     0.3  2.7     2.8     4.4     6.4     8.7    12.6
## x[1,470]          6.2     0.3  2.2     3.2     4.4     6.1     7.5    10.4
## x[1,471]          5.7     0.3  2.0     2.2     4.2     5.7     7.0     9.4
## x[1,472]          3.9     0.2  1.4     1.6     2.8     3.7     4.7     6.6
## x[1,473]          1.7     0.2  1.2     0.2     0.8     1.3     2.3     3.7
## x[1,474]          0.1     0.1  1.4    -2.3    -0.7     0.3     1.0     2.5
## x[1,475]         -1.2     0.2  1.4    -4.4    -2.1    -0.9    -0.3     1.0
## x[1,476]         -2.7     0.2  1.8    -6.3    -3.4    -2.5    -1.2    -0.3
## x[1,477]         -3.3     0.2  1.7    -7.0    -4.4    -3.4    -1.9    -0.7
## x[1,478]         -4.0     0.2  1.7    -6.8    -5.1    -4.0    -2.6    -0.8
## x[1,479]         -4.4     0.2  2.0    -8.2    -5.7    -4.2    -2.7    -1.6
## x[1,480]         -3.6     0.2  2.1    -8.2    -4.9    -3.1    -2.2    -0.8
## x[1,481]         -1.9     0.1  1.1    -4.3    -2.6    -1.8    -1.3     0.1
## x[1,482]         -1.2     0.1  1.3    -3.6    -2.1    -1.0    -0.4     0.9
## x[1,483]         -1.1     0.1  1.2    -3.2    -1.8    -1.3    -0.1     0.9
## x[1,484]         -1.2     0.2  1.3    -3.7    -2.1    -1.4    -0.7     1.3
## x[1,485]         -0.9     0.1  1.2    -3.6    -1.6    -0.6    -0.1     1.0
## x[1,486]         -0.3     0.1  1.1    -2.2    -0.9    -0.1     0.4     2.0
## x[1,487]         -0.1     0.1  1.0    -1.7    -0.8    -0.1     0.6     2.0
## x[1,488]          0.1     0.2  1.1    -1.8    -0.5     0.4     0.9     1.6
## x[1,489]          0.5     0.1  1.0    -1.3    -0.1     0.5     1.1     2.8
## x[1,490]          0.9     0.2  1.3    -0.9     0.0     0.6     1.4     4.2
## x[1,491]          1.2     0.2  1.5    -0.8     0.2     0.9     1.7     5.2
## x[1,492]          1.4     0.2  1.6    -0.7     0.3     1.2     1.9     5.6
## Z[1,1]            0.3     0.0  0.2     0.2     0.2     0.3     0.3     0.7
## Z[2,1]            0.2     0.0  0.1     0.1     0.2     0.2     0.3     0.5
## Z[3,1]            0.2     0.0  0.1     0.1     0.2     0.2     0.3     0.4
## sigma[1]          0.9     0.0  0.0     0.9     0.9     0.9     0.9     0.9
## log_lik[1]       -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[2]       -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[3]       -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[4]       -1.0     0.0  0.2    -1.4    -1.1    -0.9    -0.8    -0.8
## log_lik[5]       -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[6]       -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[7]       -1.0     0.0  0.2    -1.4    -1.1    -0.9    -0.8    -0.8
## log_lik[8]       -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[9]       -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[10]      -1.0     0.0  0.2    -1.3    -1.0    -0.9    -0.9    -0.8
## log_lik[11]      -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[12]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[13]      -1.1     0.0  0.2    -1.6    -1.2    -1.1    -1.0    -0.8
## log_lik[14]      -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[15]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[16]      -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[17]      -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[18]      -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[19]      -1.0     0.0  0.2    -1.4    -1.1    -0.9    -0.9    -0.8
## log_lik[20]      -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[21]      -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[22]      -1.1     0.0  0.3    -1.6    -1.2    -1.0    -0.9    -0.8
## log_lik[23]      -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[24]      -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[25]      -1.0     0.0  0.2    -1.5    -1.1    -0.9    -0.8    -0.8
## log_lik[26]      -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[27]      -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[28]      -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[29]      -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[30]      -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[31]      -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[32]      -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[33]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[34]      -0.9     0.0  0.2    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[35]      -0.8     0.0  0.1    -1.1    -0.8    -0.8    -0.8    -0.8
## log_lik[36]      -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[37]      -0.9     0.0  0.2    -1.6    -1.0    -0.8    -0.8    -0.8
## log_lik[38]      -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[39]      -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[40]      -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[41]      -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[42]      -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[43]      -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[44]      -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[45]      -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[46]      -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[47]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[48]      -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[49]      -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[50]      -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[51]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[52]      -1.0     0.0  0.2    -1.6    -1.1    -1.0    -0.8    -0.8
## log_lik[53]      -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[54]      -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[55]      -1.0     0.0  0.2    -1.7    -1.1    -0.9    -0.8    -0.8
## log_lik[56]      -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[57]      -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[58]      -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[59]      -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[60]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[61]      -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[62]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[63]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[64]      -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[65]      -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[66]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[67]      -1.0     0.0  0.3    -1.4    -1.1    -0.9    -0.8    -0.8
## log_lik[68]      -0.9     0.0  0.2    -1.4    -0.9    -0.8    -0.8    -0.8
## log_lik[69]      -0.9     0.0  0.2    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[70]      -1.0     0.0  0.2    -1.5    -1.2    -1.0    -0.8    -0.8
## log_lik[71]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[72]      -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[73]      -1.1     0.0  0.2    -1.5    -1.2    -1.0    -0.9    -0.8
## log_lik[74]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[75]      -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[76]      -1.0     0.0  0.2    -1.6    -1.1    -1.0    -0.9    -0.8
## log_lik[77]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[78]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[79]      -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[80]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[81]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[82]      -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[83]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[84]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[85]      -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[86]      -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[87]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[88]      -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[89]      -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[90]      -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[91]      -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[92]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[93]      -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[94]      -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[95]      -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[96]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[97]      -0.9     0.0  0.1    -1.2    -1.0    -0.8    -0.8    -0.8
## log_lik[98]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[99]      -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[100]     -0.9     0.0  0.2    -1.4    -0.9    -0.8    -0.8    -0.8
## log_lik[101]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[102]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[103]     -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[104]     -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[105]     -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[106]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[107]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[108]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[109]     -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[110]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[111]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[112]     -0.9     0.0  0.2    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[113]     -0.9     0.0  0.2    -1.4    -0.9    -0.9    -0.8    -0.8
## log_lik[114]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[115]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[116]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[117]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[118]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[119]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[120]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[121]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[122]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[123]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[124]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[125]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[126]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[127]     -0.9     0.0  0.2    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[128]     -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[129]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[130]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[131]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[132]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[133]     -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[134]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[135]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[136]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[137]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[138]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[139]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[140]     -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[141]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[142]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[143]     -1.1     0.0  0.2    -1.5    -1.2    -1.0    -0.9    -0.8
## log_lik[144]     -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[145]     -1.0     0.0  0.2    -1.6    -1.1    -0.9    -0.8    -0.8
## log_lik[146]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[147]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[148]     -0.9     0.0  0.1    -1.0    -0.9    -0.9    -0.8    -0.8
## log_lik[149]     -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[150]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[151]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[152]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[153]     -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[154]     -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[155]     -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[156]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[157]     -1.0     0.0  0.3    -1.4    -1.1    -1.0    -0.8    -0.8
## log_lik[158]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[159]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[160]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[161]     -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[162]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[163]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[164]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[165]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[166]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[167]     -0.8     0.0  0.0    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[168]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[169]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[170]     -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[171]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[172]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[173]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[174]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[175]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[176]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[177]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[178]     -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[179]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[180]     -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[181]     -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[182]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[183]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[184]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[185]     -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[186]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[187]     -0.9     0.0  0.3    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[188]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[189]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[190]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[191]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[192]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[193]     -1.0     0.0  0.2    -1.5    -1.0    -0.9    -0.8    -0.8
## log_lik[194]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[195]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[196]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[197]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[198]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[199]     -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[200]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[201]     -0.8     0.0  0.1    -1.1    -0.8    -0.8    -0.8    -0.8
## log_lik[202]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[203]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[204]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[205]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[206]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[207]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[208]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[209]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[210]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[211]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[212]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[213]     -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[214]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[215]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[216]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[217]     -1.9     0.1  0.6    -3.2    -2.3    -1.7    -1.5    -1.2
## log_lik[218]     -2.7     0.1  0.5    -3.5    -3.1    -2.8    -2.3    -1.7
## log_lik[219]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[220]     -1.5     0.1  0.5    -2.6    -1.9    -1.3    -1.1    -0.9
## log_lik[221]     -2.9     0.1  0.7    -3.9    -3.5    -3.1    -2.3    -1.4
## log_lik[222]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[223]     -1.3     NaN  0.5    -2.3    -1.6    -1.2    -0.9    -0.8
## log_lik[224]     -2.8     NaN  0.8    -4.2    -3.5    -2.9    -2.3    -1.2
## log_lik[225]     -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[226]     -1.2     0.0  0.3    -1.9    -1.3    -1.1    -1.0    -0.8
## log_lik[227]     -2.8     0.1  0.6    -3.8    -3.2    -2.8    -2.3    -1.7
## log_lik[228]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[229]     -1.0     0.0  0.2    -1.3    -1.2    -1.0    -0.9    -0.8
## log_lik[230]     -2.8     0.1  0.5    -3.9    -3.3    -2.7    -2.5    -2.0
## log_lik[231]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[232]     -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[233]     -3.0     0.1  0.7    -4.6    -3.3    -2.9    -2.5    -2.0
## log_lik[234]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[235]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[236]     -3.4     0.1  0.7    -4.6    -3.8    -3.5    -2.9    -2.3
## log_lik[237]     -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[238]     -0.8     0.0  0.1    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[239]     -4.4     0.1  0.6    -5.7    -4.9    -4.3    -4.0    -3.4
## log_lik[240]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[241]     -1.5     0.0  0.4    -2.2    -1.8    -1.5    -1.2    -1.0
## log_lik[242]    -17.7     0.2  1.7   -20.2   -19.0   -17.9   -16.6   -14.5
## log_lik[243]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[244]     -2.2     0.0  0.6    -3.4    -2.5    -2.2    -1.9    -1.1
## log_lik[245]    -11.7     0.2  1.7   -14.6   -12.7   -12.0   -10.4    -9.1
## log_lik[246]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[247]     -1.6     0.1  0.5    -2.6    -1.9    -1.5    -1.3    -1.0
## log_lik[248]     -8.3     0.1  1.4   -10.5    -9.2    -8.4    -7.7    -5.9
## log_lik[249]     -0.9     0.0  0.1    -1.3    -1.0    -0.8    -0.8    -0.8
## log_lik[250]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[251]     -7.2     0.1  1.3    -9.7    -8.2    -7.2    -6.3    -4.7
## log_lik[252]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[253]     -3.0     0.1  0.6    -4.1    -3.5    -3.0    -2.7    -2.0
## log_lik[254]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[255]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[256]     -1.5     0.1  0.4    -2.5    -1.8    -1.4    -1.3    -1.0
## log_lik[257]     -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[258]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[259]     -1.0     0.0  0.2    -1.5    -1.1    -0.9    -0.8    -0.8
## log_lik[260]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[261]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[262]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[263]     -0.8     0.0  0.1    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[264]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[265]     -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[266]     -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[267]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[268]     -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[269]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[270]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[271]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[272]     -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[273]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[274]     -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[275]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[276]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[277]     -0.9     0.0  0.2    -1.4    -0.9    -0.8    -0.8    -0.8
## log_lik[278]     -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[279]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[280]     -1.3     0.0  0.4    -2.4    -1.5    -1.1    -1.0    -0.8
## log_lik[281]     -1.0     0.0  0.1    -1.3    -1.0    -1.0    -0.8    -0.8
## log_lik[282]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[283]     -1.6     0.1  0.5    -2.9    -1.9    -1.6    -1.2    -0.9
## log_lik[284]     -1.0     0.0  0.2    -1.4    -1.2    -0.9    -0.9    -0.8
## log_lik[285]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[286]     -1.2     0.1  0.4    -2.0    -1.3    -1.2    -1.0    -0.8
## log_lik[287]     -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[288]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[289]     -1.4     0.0  0.4    -2.3    -1.7    -1.4    -1.1    -0.9
## log_lik[290]     -1.0     0.0  0.2    -1.3    -1.1    -1.0    -0.9    -0.8
## log_lik[291]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[292]     -1.1     0.0  0.2    -1.5    -1.3    -1.1    -0.9    -0.8
## log_lik[293]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[294]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[295]     -1.0     0.0  0.2    -1.4    -1.1    -0.9    -0.8    -0.8
## log_lik[296]     -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
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## log_lik[471]     -1.0     0.0  0.1    -1.2    -1.0    -1.0    -0.8    -0.8
## log_lik[472]     -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[473]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[474]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[475]     -1.0     0.0  0.2    -1.6    -1.1    -1.0    -0.8    -0.8
## log_lik[476]     -1.5     0.0  0.3    -2.4    -1.7    -1.4    -1.3    -1.0
## log_lik[477]     -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[478]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[479]     -0.8     0.0  0.1    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[480]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[481]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[482]     -1.1     0.0  0.2    -1.4    -1.2    -1.1    -1.0    -0.8
## log_lik[483]     -0.8     0.0  0.1    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[484]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[485]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[486]     -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[487]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[488]     -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[489]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[490]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[491]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.9    -0.8
## log_lik[492]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[493]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[494]     -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[495]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[496]     -0.9     0.0  0.2    -1.4    -0.9    -0.8    -0.8    -0.8
## log_lik[497]     -1.1     0.1  0.3    -1.8    -1.2    -1.0    -0.9    -0.8
## log_lik[498]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[499]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[500]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[501]     -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[502]     -1.0     0.0  0.3    -1.8    -1.1    -1.0    -0.8    -0.8
## log_lik[503]     -1.4     0.0  0.3    -2.0    -1.6    -1.3    -1.2    -0.9
## log_lik[504]     -1.2     0.0  0.2    -1.8    -1.3    -1.2    -1.0    -0.8
## log_lik[505]     -1.8     0.0  0.6    -3.3    -2.2    -1.8    -1.4    -1.0
## log_lik[506]     -2.5     0.1  0.7    -3.7    -2.9    -2.4    -2.0    -1.4
## log_lik[507]     -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[508]     -1.8     0.1  0.6    -3.1    -2.1    -1.5    -1.3    -1.1
## log_lik[509]     -1.3     0.0  0.3    -1.7    -1.6    -1.3    -1.0    -0.8
## log_lik[510]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[511]     -1.1     0.0  0.2    -1.4    -1.3    -1.0    -0.8    -0.8
## log_lik[512]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[513]     -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[514]     -0.9     0.0  0.2    -1.4    -1.0    -0.8    -0.8    -0.8
## log_lik[515]     -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[516]     -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[517]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[518]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[519]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[520]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[521]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[522]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[523]     -1.1     0.0  0.3    -1.8    -1.3    -1.0    -0.9    -0.8
## log_lik[524]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[525]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[526]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[527]     -1.1     0.0  0.2    -1.4    -1.2    -1.1    -1.0    -0.9
## log_lik[528]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[529]     -1.0     0.0  0.2    -1.7    -1.1    -0.9    -0.9    -0.8
## log_lik[530]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[531]     -0.8     0.0  0.1    -1.1    -0.8    -0.8    -0.8    -0.8
## log_lik[532]     -1.2     0.0  0.3    -1.8    -1.3    -1.1    -1.0    -0.8
## log_lik[533]     -1.1     0.0  0.2    -1.6    -1.2    -1.1    -1.0    -0.9
## log_lik[534]     -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.9    -0.8
## log_lik[535]     -1.0     0.0  0.2    -1.5    -1.1    -0.9    -0.9    -0.8
## log_lik[536]     -0.9     0.0  0.1    -1.0    -0.9    -0.9    -0.8    -0.8
## log_lik[537]     -1.0     0.0  0.2    -1.4    -1.0    -0.9    -0.9    -0.8
## log_lik[538]     -1.3     0.0  0.4    -2.0    -1.6    -1.3    -1.0    -0.9
## log_lik[539]     -1.2     0.0  0.2    -1.7    -1.4    -1.2    -1.0    -0.9
## log_lik[540]     -1.4     0.0  0.3    -2.1    -1.6    -1.3    -1.1    -1.0
## log_lik[541]     -1.2     0.0  0.3    -1.7    -1.3    -1.1    -0.9    -0.8
## log_lik[542]     -1.1     0.0  0.2    -1.5    -1.2    -1.1    -0.9    -0.8
## log_lik[543]     -1.1     0.0  0.2    -1.4    -1.2    -1.1    -0.9    -0.8
## log_lik[544]     -0.9     0.0  0.1    -1.2    -1.0    -0.8    -0.8    -0.8
## log_lik[545]     -1.6     0.0  0.4    -2.1    -1.8    -1.6    -1.3    -1.0
## log_lik[546]     -1.0     0.0  0.1    -1.2    -1.1    -1.0    -0.9    -0.8
## log_lik[547]     -1.9     0.1  0.4    -2.5    -2.1    -1.9    -1.6    -1.2
## log_lik[548]     -1.3     0.0  0.3    -1.7    -1.4    -1.2    -1.1    -0.9
## log_lik[549]     -1.1     0.0  0.2    -1.5    -1.2    -1.1    -1.0    -0.9
## log_lik[550]     -1.2     0.0  0.3    -1.9    -1.3    -1.1    -1.0    -0.9
## log_lik[551]     -1.7     0.1  0.4    -2.3    -1.9    -1.7    -1.5    -1.0
## log_lik[552]     -1.3     0.0  0.3    -1.8    -1.5    -1.3    -1.1    -0.9
## log_lik[553]     -1.6     0.1  0.5    -3.0    -1.9    -1.5    -1.3    -1.0
## log_lik[554]     -1.3     0.0  0.3    -1.8    -1.5    -1.3    -1.1    -0.8
## log_lik[555]     -1.2     0.0  0.2    -1.7    -1.3    -1.2    -1.0    -0.8
## log_lik[556]     -1.1     0.0  0.3    -1.8    -1.2    -1.1    -0.9    -0.8
## log_lik[557]     -1.7     0.1  0.4    -2.4    -1.9    -1.7    -1.6    -1.1
## log_lik[558]     -1.5     0.0  0.3    -2.0    -1.7    -1.4    -1.3    -0.9
## log_lik[559]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[560]     -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[561]     -2.1     0.1  0.5    -3.0    -2.4    -2.1    -1.7    -1.3
## log_lik[562]     -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[563]     -2.7     0.1  0.6    -4.0    -3.1    -2.7    -2.3    -1.9
## log_lik[564]     -2.8     0.0  0.6    -4.0    -3.1    -2.8    -2.4    -1.9
## log_lik[565]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[566]     -2.0     0.0  0.4    -2.8    -2.3    -2.0    -1.8    -1.3
## log_lik[567]     -4.9     0.1  0.7    -6.5    -5.2    -4.8    -4.5    -3.7
## log_lik[568]     -1.5     0.1  0.6    -3.0    -1.8    -1.4    -1.2    -0.9
## log_lik[569]     -1.3     0.1  0.3    -2.0    -1.5    -1.3    -1.1    -0.8
## log_lik[570]     -9.4     0.2  1.5   -11.8   -10.3    -9.7    -8.2    -6.7
## log_lik[571]     -1.4     0.1  0.4    -2.2    -1.6    -1.4    -1.2    -0.9
## log_lik[572]     -1.6     0.0  0.4    -2.2    -1.9    -1.6    -1.3    -1.1
## log_lik[573]    -10.7     0.2  1.4   -12.8   -11.5   -10.9    -9.6    -8.4
## log_lik[574]     -0.9     0.0  0.2    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[575]     -2.7     0.1  0.6    -3.8    -3.1    -2.7    -2.3    -1.8
## log_lik[576]     -6.8     0.1  1.2    -8.7    -7.6    -7.0    -6.1    -4.8
## log_lik[577]     -1.5     0.1  0.5    -2.7    -1.7    -1.4    -1.1    -0.8
## log_lik[578]     -5.5     0.1  1.2    -7.9    -6.0    -5.4    -4.9    -3.7
## log_lik[579]     -2.7     0.1  0.7    -4.0    -3.2    -2.6    -2.4    -1.8
## log_lik[580]     -1.4     0.1  0.5    -2.6    -1.8    -1.2    -1.1    -0.9
## log_lik[581]     -3.2     0.1  0.7    -4.1    -3.7    -3.4    -2.7    -1.9
## log_lik[582]     -1.2     0.0  0.3    -1.7    -1.4    -1.3    -1.0    -0.8
## log_lik[583]     -1.2     0.0  0.3    -1.9    -1.4    -1.2    -0.9    -0.8
## log_lik[584]     -1.5     0.1  0.5    -2.5    -1.7    -1.4    -1.2    -0.9
## log_lik[585]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[586]     -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[587]     -1.2     0.1  0.3    -2.1    -1.2    -1.1    -1.0    -0.9
## log_lik[588]     -0.9     0.0  0.2    -1.5    -1.0    -0.9    -0.8    -0.8
## log_lik[589]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[590]     -1.0     0.0  0.2    -1.3    -1.1    -0.9    -0.8    -0.8
## log_lik[591]     -1.0     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[592]     -1.0     0.0  0.3    -1.6    -1.1    -1.0    -0.8    -0.8
## log_lik[593]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[594]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[595]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[596]     -0.9     0.0  0.1    -1.1    -1.0    -0.8    -0.8    -0.8
## log_lik[597]     -2.7     NaN  0.5    -3.6    -3.1    -2.7    -2.3    -1.8
## log_lik[598]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[599]     -1.5     0.0  0.3    -2.2    -1.8    -1.4    -1.2    -1.0
## log_lik[600]     -9.5     0.1  1.2   -11.1   -10.5    -9.7    -8.6    -7.5
## log_lik[601]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[602]     -2.1     0.0  0.4    -3.0    -2.3    -2.0    -1.8    -1.5
## log_lik[603]     -5.0     0.1  0.7    -6.0    -5.4    -5.1    -4.7    -3.4
## log_lik[604]     -1.6     0.1  0.5    -2.8    -1.8    -1.6    -1.3    -1.1
## log_lik[605]     -1.7     0.1  0.4    -2.4    -2.1    -1.6    -1.4    -1.1
## log_lik[606]     -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[607]     -3.2     0.1  0.8    -4.9    -3.8    -3.2    -2.6    -2.0
## log_lik[608]     -1.1     0.0  0.3    -1.7    -1.4    -1.1    -0.9    -0.8
## log_lik[609]    -17.6     0.4  2.4   -21.6   -19.3   -17.9   -16.0   -13.2
## log_lik[610]     -4.7     0.1  1.3    -7.4    -5.4    -4.7    -3.8    -2.8
## log_lik[611]     -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[612]    -13.9     0.2  2.5   -18.0   -16.0   -13.8   -12.5    -9.8
## log_lik[613]     -1.4     0.1  0.6    -3.0    -1.5    -1.2    -1.0    -0.8
## log_lik[614]     -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[615]    -25.0     0.2  3.9   -31.7   -27.6   -25.2   -21.9   -18.5
## log_lik[616]     -1.8     0.1  0.7    -3.3    -2.1    -1.7    -1.2    -0.9
## log_lik[617]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[618]     -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[619]     -2.1     NaN  0.8    -3.7    -2.6    -1.9    -1.5    -1.2
## log_lik[620]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[621]     -0.9     0.0  0.2    -1.5    -1.0    -0.8    -0.8    -0.8
## log_lik[622]     -2.3     0.0  0.8    -3.9    -2.6    -2.2    -1.6    -1.2
## log_lik[623]     -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[624]    -68.0     0.3  5.3   -74.8   -73.2   -68.4   -64.0   -59.5
## log_lik[625]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[626]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[627]     -3.3     0.1  0.7    -4.8    -3.5    -3.3    -2.7    -2.3
## log_lik[628]     -0.9     0.0  0.2    -1.3    -1.0    -0.8    -0.8    -0.8
## log_lik[629]     -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[630]     -3.1     0.1  0.5    -4.1    -3.4    -3.2    -2.8    -2.2
## log_lik[631]     -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[632]     -0.9     0.0  0.2    -1.5    -0.9    -0.9    -0.8    -0.8
## log_lik[633]     -2.8     0.1  0.6    -3.8    -3.0    -2.8    -2.5    -1.4
## log_lik[634]     -1.2     0.1  0.4    -2.0    -1.4    -1.1    -0.9    -0.8
## log_lik[635]     -0.8     0.0  0.1    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[636]     -2.6     0.1  0.6    -3.6    -3.1    -2.7    -2.1    -1.6
## log_lik[637]     -1.3     0.0  0.4    -2.3    -1.4    -1.1    -1.0    -0.8
## log_lik[638]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[639]     -3.7     0.1  0.8    -4.8    -4.3    -3.8    -3.1    -2.2
## log_lik[640]     -1.2     0.0  0.3    -1.8    -1.4    -1.1    -0.9    -0.8
## log_lik[641]     -1.0     0.0  0.2    -1.6    -1.1    -0.9    -0.8    -0.8
## log_lik[642]     -3.9     0.2  1.0    -5.8    -4.7    -3.9    -3.1    -2.5
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## log_lik[687]     -1.3     0.0  0.3    -1.8    -1.5    -1.2    -1.1    -0.9
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## log_lik[700]     -0.9     0.0  0.2    -1.4    -0.9    -0.9    -0.8    -0.8
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## log_lik[702]     -1.0     0.0  0.1    -1.3    -1.0    -0.9    -0.9    -0.8
## log_lik[703]     -0.9     0.0  0.2    -1.3    -0.9    -0.9    -0.8    -0.8
## log_lik[704]     -0.9     0.0  0.1    -1.2    -0.8    -0.8    -0.8    -0.8
## log_lik[705]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[706]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[707]     -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[708]     -0.9     0.0  0.1    -1.0    -0.9    -0.9    -0.8    -0.8
## log_lik[709]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[710]     -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[711]     -0.9     0.0  0.1    -1.0    -0.9    -0.9    -0.8    -0.8
## log_lik[712]     -0.9     0.0  0.2    -1.3    -1.0    -0.8    -0.8    -0.8
## log_lik[713]     -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[714]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[715]     -0.9     0.0  0.2    -1.6    -0.9    -0.9    -0.8    -0.8
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## log_lik[717]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
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## log_lik[724]     -1.1     0.1  0.3    -1.8    -1.2    -1.1    -0.9    -0.8
## log_lik[725]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
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## log_lik[728]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[729]     -1.2     0.0  0.2    -1.8    -1.3    -1.2    -1.0    -0.9
## log_lik[730]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
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## log_lik[757]     -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
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## log_lik[759]     -1.2     0.0  0.2    -1.7    -1.4    -1.2    -1.1    -0.9
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## log_lik[761]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
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## log_lik[812]     -0.8     0.0  0.1    -1.1    -0.8    -0.8    -0.8    -0.8
## log_lik[813]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[814]     -1.0     0.0  0.2    -1.3    -1.1    -0.9    -0.9    -0.8
## log_lik[815]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[816]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[817]     -0.9     0.0  0.2    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[818]     -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[819]     -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[820]     -1.3     0.0  0.3    -1.8    -1.5    -1.3    -1.1    -0.9
## log_lik[821]     -1.0     0.0  0.1    -1.2    -1.0    -0.9    -0.9    -0.8
## log_lik[822]     -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.9    -0.8
## log_lik[823]     -1.8     0.1  0.5    -2.9    -2.1    -1.8    -1.5    -1.1
## log_lik[824]     -1.1     0.0  0.2    -1.6    -1.2    -1.1    -0.9    -0.8
## log_lik[825]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[826]     -1.9     0.1  0.6    -2.8    -2.5    -1.8    -1.4    -1.0
## log_lik[827]     -1.4     0.0  0.4    -2.1    -1.5    -1.3    -1.1    -1.0
## log_lik[828]     -1.5     0.0  0.4    -2.1    -1.8    -1.4    -1.2    -0.9
## log_lik[829]     -1.6     0.1  0.4    -2.5    -1.9    -1.6    -1.3    -1.0
## log_lik[830]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[831]     -2.2     0.1  0.5    -3.1    -2.5    -2.3    -1.8    -1.6
## log_lik[832]     -2.1     0.1  0.5    -3.3    -2.4    -2.0    -1.7    -1.4
## log_lik[833]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[834]     -3.1     0.1  0.5    -3.8    -3.5    -3.1    -2.8    -2.1
## log_lik[835]     -2.4     0.1  0.7    -4.1    -2.7    -2.2    -1.9    -1.4
## log_lik[836]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[837]     -3.5     0.1  0.7    -4.6    -3.9    -3.6    -3.2    -2.1
## log_lik[838]     -1.4     0.1  0.4    -2.2    -1.7    -1.4    -1.2    -0.9
## log_lik[839]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[840]     -2.7     0.1  0.5    -3.7    -3.0    -2.6    -2.4    -1.9
## log_lik[841]     -1.2     0.0  0.3    -1.9    -1.4    -1.1    -0.9    -0.8
## log_lik[842]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[843]     -1.9     0.1  0.5    -3.0    -2.2    -1.9    -1.5    -1.3
## log_lik[844]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[845]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[846]     -1.1     0.0  0.3    -1.8    -1.1    -1.1    -1.0    -0.8
## log_lik[847]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[848]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[849]     -0.9     0.0  0.1    -1.0    -0.9    -0.9    -0.8    -0.8
## log_lik[850]     -1.2     0.1  0.3    -1.9    -1.5    -1.2    -0.9    -0.8
## log_lik[851]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[852]     -1.9     0.1  0.4    -2.7    -2.2    -1.8    -1.5    -1.4
## log_lik[853]     -2.0     0.1  0.6    -3.1    -2.3    -1.8    -1.6    -1.1
## log_lik[854]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[855]     -3.6     0.1  0.6    -4.8    -4.0    -3.6    -3.3    -2.4
## log_lik[856]     -2.5     0.0  0.6    -3.4    -2.9    -2.5    -2.1    -1.3
## log_lik[857]     -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[858]     -5.0     0.0  0.7    -6.4    -5.4    -5.0    -4.5    -3.9
## log_lik[859]     -1.7     0.1  0.5    -2.8    -2.1    -1.7    -1.4    -0.9
## log_lik[860]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[861]     -3.5     0.1  0.7    -4.8    -3.9    -3.5    -3.0    -2.5
## log_lik[862]     -2.4     0.1  0.5    -3.3    -2.7    -2.4    -2.0    -1.5
## log_lik[863]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[864]     -1.1     0.0  0.2    -1.5    -1.3    -1.1    -1.0    -0.9
## log_lik[865]     -2.9     0.1  0.7    -4.6    -3.2    -2.7    -2.5    -1.8
## log_lik[866]     -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[867]     -1.5     0.0  0.3    -2.3    -1.7    -1.5    -1.4    -1.0
## log_lik[868]     -1.2     0.1  0.3    -1.9    -1.4    -1.1    -0.9    -0.8
## log_lik[869]     -1.1     0.0  0.2    -1.5    -1.2    -1.0    -0.9    -0.8
## log_lik[870]     -1.0     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[871]     -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[872]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[873]     -1.0     0.0  0.1    -1.3    -1.1    -0.9    -0.9    -0.8
## log_lik[874]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[875]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[876]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[877]     -1.0     0.0  0.2    -1.5    -1.1    -0.9    -0.8    -0.8
## log_lik[878]     -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[879]     -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[880]     -1.0     0.0  0.2    -1.4    -1.2    -1.0    -0.8    -0.8
## log_lik[881]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[882]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[883]     -1.0     0.0  0.2    -1.5    -1.0    -0.9    -0.8    -0.8
## log_lik[884]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[885]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[886]     -1.0     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[887]     -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[888]     -0.9     0.0  0.1    -1.0    -0.9    -0.9    -0.8    -0.8
## log_lik[889]     -0.9     0.0  0.2    -1.4    -0.9    -0.8    -0.8    -0.8
## log_lik[890]     -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[891]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[892]     -0.9     0.0  0.1    -1.2    -1.0    -0.8    -0.8    -0.8
## log_lik[893]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[894]     -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[895]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[896]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[897]     -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[898]     -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[899]     -2.1     0.0  0.4    -2.9    -2.4    -2.0    -1.7    -1.5
## log_lik[900]     -1.2     0.0  0.2    -1.6    -1.4    -1.2    -1.1    -0.9
## log_lik[901]     -1.1     0.1  0.3    -1.8    -1.1    -1.0    -0.8    -0.8
## log_lik[902]     -1.2     0.1  0.3    -1.8    -1.3    -1.1    -0.9    -0.8
## log_lik[903]     -1.5     0.1  0.4    -2.3    -1.7    -1.4    -1.3    -1.0
## log_lik[904]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[905]     -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[906]     -1.1     0.0  0.3    -1.8    -1.3    -1.1    -1.0    -0.8
## log_lik[907]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[908]     -5.8     0.1  0.9    -7.4    -6.4    -5.8    -5.2    -4.2
## log_lik[909]     -1.1     0.0  0.2    -1.5    -1.2    -1.1    -1.0    -0.8
## log_lik[910]     -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[911]     -0.9     0.0  0.2    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[912]     -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[913]     -1.0     0.1  0.3    -2.0    -1.1    -1.0    -0.8    -0.8
## log_lik[914]     -1.8     0.1  0.5    -3.1    -2.0    -1.7    -1.4    -1.0
## log_lik[915]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[916]     -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[917]     -1.3     0.0  0.3    -1.8    -1.4    -1.2    -1.1    -0.8
## log_lik[918]     -1.1     0.0  0.2    -1.6    -1.3    -1.1    -1.0    -0.8
## log_lik[919]     -0.9     0.0  0.2    -1.4    -0.9    -0.9    -0.8    -0.8
## log_lik[920]     -1.9     NaN  0.5    -2.8    -2.3    -1.9    -1.5    -1.2
## log_lik[921]     -1.3     0.0  0.3    -1.9    -1.4    -1.2    -1.1    -0.8
## log_lik[922]     -0.9     0.0  0.2    -1.3    -1.0    -0.8    -0.8    -0.8
## log_lik[923]     -2.1     NaN  0.6    -3.2    -2.5    -2.0    -1.6    -1.1
## log_lik[924]     -1.2     0.0  0.3    -1.8    -1.4    -1.2    -1.0    -0.8
## log_lik[925]     -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[926]     -2.0     0.0  0.4    -2.8    -2.3    -2.1    -1.6    -1.3
## log_lik[927]     -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[928]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[929]     -1.1     0.0  0.2    -1.6    -1.3    -1.1    -1.0    -0.8
## log_lik[930]     -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[931]     -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[932]     -1.8     0.1  0.3    -2.6    -2.0    -1.8    -1.6    -1.3
## log_lik[933]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[934]     -0.9     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[935]     -1.0     0.0  0.1    -1.3    -1.0    -1.0    -0.9    -0.8
## log_lik[936]     -1.0     0.0  0.2    -1.5    -1.2    -1.0    -0.9    -0.8
## log_lik[937]     -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[938]     -1.9     0.0  0.4    -2.7    -2.1    -1.9    -1.6    -1.1
## log_lik[939]     -1.1     0.0  0.2    -1.8    -1.1    -1.0    -0.9    -0.8
## log_lik[940]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[941]     -1.3     0.0  0.3    -1.8    -1.5    -1.3    -1.1    -1.0
## log_lik[942]     -1.2     0.0  0.2    -1.6    -1.3    -1.1    -1.0    -0.9
## log_lik[943]     -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[944]     -1.2     0.0  0.2    -1.6    -1.3    -1.2    -1.0    -0.8
## log_lik[945]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[946]     -0.9     0.0  0.1    -1.3    -0.9    -0.9    -0.8    -0.8
## log_lik[947]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[948]     -1.0     0.0  0.1    -1.3    -1.0    -0.9    -0.9    -0.8
## log_lik[949]     -0.9     0.0  0.2    -1.4    -0.9    -0.8    -0.8    -0.8
## log_lik[950]     -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[951]     -0.9     0.0  0.1    -1.0    -1.0    -0.9    -0.8    -0.8
## log_lik[952]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[953]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[954]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[955]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[956]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[957]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[958]     -0.9     0.0  0.2    -1.4    -0.9    -0.9    -0.8    -0.8
## log_lik[959]     -1.0     0.0  0.1    -1.3    -1.1    -1.0    -1.0    -0.8
## log_lik[960]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[961]     -1.0     0.0  0.2    -1.6    -1.0    -0.9    -0.8    -0.8
## log_lik[962]     -1.1     0.0  0.2    -1.6    -1.2    -1.2    -1.0    -0.8
## log_lik[963]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[964]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[965]     -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[966]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[967]     -0.9     0.0  0.2    -1.5    -1.0    -0.9    -0.8    -0.8
## log_lik[968]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[969]     -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[970]     -1.0     0.0  0.3    -1.7    -1.1    -0.9    -0.8    -0.8
## log_lik[971]     -1.2     0.1  0.3    -2.1    -1.3    -1.1    -1.0    -0.8
## log_lik[972]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[973]     -1.6     0.1  0.5    -2.7    -1.6    -1.5    -1.3    -1.0
## log_lik[974]     -1.3     0.0  0.2    -1.8    -1.4    -1.2    -1.1    -0.9
## log_lik[975]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.9    -0.8
## log_lik[976]     -1.4     0.1  0.4    -2.3    -1.6    -1.2    -1.0    -0.8
## log_lik[977]     -1.0     0.0  0.2    -1.3    -1.1    -0.9    -0.8    -0.8
## log_lik[978]     -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[979]     -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[980]     -1.1     0.0  0.2    -1.4    -1.2    -1.0    -0.9    -0.8
## log_lik[981]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[982]     -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[983]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
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## log_lik[985]     -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[986]     -0.8     0.0  0.1    -0.9    -0.9    -0.8    -0.8    -0.8
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## log_lik[988]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[989]     -0.8     0.0  0.1    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[990]     -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[991]     -0.9     0.0  0.1    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[992]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[993]     -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[994]     -0.9     0.0  0.2    -1.4    -0.9    -0.9    -0.8    -0.8
## log_lik[995]     -0.9     0.0  0.2    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[996]     -0.8     0.0  0.1    -1.1    -0.8    -0.8    -0.8    -0.8
## log_lik[997]     -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[998]     -1.0     0.0  0.1    -1.2    -1.0    -1.0    -0.9    -0.8
## log_lik[999]     -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1000]    -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[1001]    -1.0     0.0  0.2    -1.3    -1.1    -1.0    -0.9    -0.8
## log_lik[1002]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1003]    -1.7     0.1  0.6    -2.8    -1.9    -1.6    -1.3    -1.0
## log_lik[1004]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1005]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1006]    -2.8     0.1  0.8    -4.4    -3.3    -2.8    -2.3    -1.7
## log_lik[1007]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1008]    -0.8     0.0  0.1    -1.1    -0.8    -0.8    -0.8    -0.8
## log_lik[1009]    -1.4     0.0  0.4    -2.2    -1.6    -1.3    -1.0    -0.9
## log_lik[1010]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1011]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1012]    -1.4     0.1  0.5    -2.5    -1.7    -1.4    -1.0    -0.8
## log_lik[1013]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1014]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1015]    -2.0     0.1  0.7    -3.7    -2.4    -1.9    -1.7    -1.0
## log_lik[1016]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1017]    -0.9     0.0  0.2    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[1018]    -1.4     0.1  0.3    -2.1    -1.6    -1.3    -1.2    -0.9
## log_lik[1019]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1020]    -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[1021]    -1.2     0.0  0.3    -2.0    -1.4    -1.2    -0.9    -0.8
## log_lik[1022]    -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[1023]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1024]    -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[1025]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1026]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.9    -0.8
## log_lik[1027]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1028]    -1.0     0.0  0.2    -1.5    -1.0    -0.9    -0.9    -0.8
## log_lik[1029]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1030]    -1.7     0.1  0.5    -2.7    -1.9    -1.6    -1.3    -1.0
## log_lik[1031]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1032]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1033]    -1.0     0.0  0.3    -1.6    -1.2    -0.9    -0.8    -0.8
## log_lik[1034]    -1.3     0.0  0.3    -1.8    -1.4    -1.2    -1.0    -0.8
## log_lik[1035]    -1.0     0.0  0.2    -1.4    -1.1    -0.9    -0.8    -0.8
## log_lik[1036]    -1.5     0.0  0.4    -2.2    -1.9    -1.5    -1.2    -0.9
## log_lik[1037]    -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[1038]    -1.0     0.0  0.2    -1.4    -1.1    -0.9    -0.8    -0.8
## log_lik[1039]    -1.3     0.1  0.5    -2.5    -1.4    -1.2    -1.0    -0.8
## log_lik[1040]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1041]    -0.9     0.0  0.1    -1.2    -1.0    -0.8    -0.8    -0.8
## log_lik[1042]    -1.7     0.1  0.6    -2.7    -2.1    -1.5    -1.2    -1.0
## log_lik[1043]    -1.1     0.0  0.2    -1.6    -1.2    -1.0    -0.9    -0.8
## log_lik[1044]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1045]    -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[1046]    -1.1     0.0  0.3    -1.7    -1.3    -1.1    -1.0    -0.8
## log_lik[1047]    -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1048]    -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[1049]    -1.0     0.0  0.2    -1.6    -1.1    -1.0    -0.9    -0.8
## log_lik[1050]    -1.0     0.0  0.2    -1.3    -1.0    -0.9    -0.9    -0.8
## log_lik[1051]    -1.1     0.0  0.3    -1.8    -1.1    -1.0    -0.9    -0.8
## log_lik[1052]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1053]    -1.0     0.0  0.1    -1.3    -1.1    -1.0    -0.9    -0.8
## log_lik[1054]    -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[1055]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1056]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1057]    -1.3     0.0  0.4    -2.2    -1.5    -1.3    -1.0    -0.8
## log_lik[1058]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1059]    -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1060]    -1.1     0.0  0.2    -1.6    -1.2    -0.9    -0.9    -0.8
## log_lik[1061]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1062]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1063]    -0.9     0.0  0.1    -1.0    -0.9    -0.9    -0.8    -0.8
## log_lik[1064]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1065]    -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[1066]    -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[1067]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1068]    -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1069]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1070]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1071]    -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[1072]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1073]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1074]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1075]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1076]    -0.9     0.0  0.1    -1.0    -0.9    -0.9    -0.8    -0.8
## log_lik[1077]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1078]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1079]    -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[1080]    -0.9     0.0  0.1    -1.4    -0.9    -0.9    -0.8    -0.8
## log_lik[1081]    -1.2     0.0  0.2    -1.6    -1.3    -1.2    -1.0    -0.8
## log_lik[1082]    -1.0     0.0  0.1    -1.3    -1.1    -0.9    -0.9    -0.8
## log_lik[1083]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1084]    -1.1     0.0  0.2    -1.6    -1.3    -1.1    -0.9    -0.8
## log_lik[1085]    -1.0     0.0  0.2    -1.4    -1.0    -0.9    -0.9    -0.8
## log_lik[1086]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1087]    -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[1088]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1089]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1090]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1091]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1092]    -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[1093]    -1.1     0.0  0.3    -1.6    -1.3    -1.0    -0.8    -0.8
## log_lik[1094]    -0.9     0.0  0.2    -1.4    -0.9    -0.8    -0.8    -0.8
## log_lik[1095]    -0.9     0.0  0.1    -1.3    -0.9    -0.9    -0.8    -0.8
## log_lik[1096]    -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[1097]    -1.1     0.0  0.2    -1.4    -1.2    -1.0    -0.9    -0.8
## log_lik[1098]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1099]    -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[1100]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1101]    -0.8     0.0  0.0    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1102]    -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[1103]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1104]    -0.8     0.0  0.1    -1.1    -0.8    -0.8    -0.8    -0.8
## log_lik[1105]    -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1106]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1107]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1108]    -1.1     0.0  0.3    -1.8    -1.2    -1.0    -0.9    -0.8
## log_lik[1109]    -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[1110]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1111]    -1.9     0.0  0.5    -2.9    -2.2    -1.7    -1.5    -1.1
## log_lik[1112]    -1.0     0.0  0.1    -1.3    -1.0    -1.0    -0.9    -0.8
## log_lik[1113]    -0.9     0.0  0.1    -1.0    -0.9    -0.9    -0.8    -0.8
## log_lik[1114]    -1.1     0.0  0.3    -2.1    -1.2    -1.0    -0.9    -0.8
## log_lik[1115]    -1.1     0.0  0.2    -1.4    -1.2    -1.1    -1.0    -0.8
## log_lik[1116]    -1.7     0.0  0.4    -2.6    -1.9    -1.7    -1.5    -1.2
## log_lik[1117]    -2.2     0.1  0.5    -3.0    -2.4    -2.1    -1.8    -1.2
## log_lik[1118]    -1.0     0.0  0.2    -1.5    -1.1    -1.0    -0.9    -0.8
## log_lik[1119]    -1.5     0.0  0.3    -2.1    -1.6    -1.5    -1.3    -1.1
## log_lik[1120]    -1.3     0.1  0.4    -2.2    -1.4    -1.2    -1.0    -0.8
## log_lik[1121]    -1.1     0.0  0.3    -1.8    -1.2    -1.0    -0.9    -0.8
## log_lik[1122]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1123]    -0.9     0.0  0.1    -1.3    -1.1    -0.9    -0.8    -0.8
## log_lik[1124]    -1.0     0.0  0.1    -1.3    -1.0    -0.9    -0.9    -0.8
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## log_lik[1126]    -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[1127]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
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## log_lik[1129]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
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## log_lik[1151]    -1.1     0.0  0.2    -1.6    -1.3    -1.0    -0.9    -0.9
## log_lik[1152]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1153]    -1.8     0.0  0.6    -3.3    -2.2    -1.7    -1.4    -1.1
## log_lik[1154]    -1.2     0.0  0.3    -1.8    -1.4    -1.1    -1.0    -0.8
## log_lik[1155]    -1.1     0.0  0.2    -1.4    -1.2    -1.0    -0.9    -0.8
## log_lik[1156]    -1.4     0.1  0.4    -2.1    -1.6    -1.4    -1.1    -1.0
## log_lik[1157]    -1.2     0.0  0.2    -1.5    -1.3    -1.2    -1.0    -0.9
## log_lik[1158]    -1.1     0.0  0.2    -1.4    -1.2    -1.1    -1.0    -0.8
## log_lik[1159]    -1.2     0.0  0.3    -1.9    -1.4    -1.2    -1.0    -0.8
## log_lik[1160]    -1.1     0.0  0.2    -1.6    -1.2    -1.0    -1.0    -0.8
## log_lik[1161]    -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[1162]    -1.1     0.0  0.2    -1.6    -1.2    -1.0    -0.9    -0.8
## log_lik[1163]    -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[1164]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1165]    -1.0     0.0  0.2    -1.6    -1.0    -0.9    -0.8    -0.8
## log_lik[1166]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1167]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1168]    -1.5     0.0  0.4    -2.5    -1.7    -1.4    -1.2    -1.0
## log_lik[1169]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1170]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1171]    -1.3     0.0  0.4    -2.2    -1.6    -1.2    -1.0    -0.8
## log_lik[1172]    -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[1173]    -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[1174]    -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[1175]    -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[1176]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1177]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1178]    -0.9     NaN  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1179]    -2.5     NaN  0.5    -3.3    -2.9    -2.5    -2.2    -1.8
## log_lik[1180]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1181]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1182]    -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1183]    -1.5     0.1  0.4    -2.1    -1.8    -1.5    -1.1    -0.9
## log_lik[1184]    -0.9     0.0  0.1    -1.1    -1.0    -0.8    -0.8    -0.8
## log_lik[1185]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1186]    -1.5     0.1  0.4    -2.3    -1.8    -1.4    -1.2    -1.0
## log_lik[1187]    -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[1188]    -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[1189]    -7.8     0.3  1.8   -10.9    -8.8    -7.9    -6.7    -4.1
## log_lik[1190]    -1.3     0.1  0.3    -2.0    -1.5    -1.2    -1.1    -0.9
## log_lik[1191]    -1.3     0.0  0.3    -1.9    -1.5    -1.2    -1.0    -0.9
## log_lik[1192]    -4.1     0.2  1.1    -5.7    -4.9    -4.4    -3.0    -2.4
## log_lik[1193]    -1.0     0.0  0.2    -1.4    -1.1    -0.9    -0.9    -0.8
## log_lik[1194]    -1.1     0.0  0.2    -1.6    -1.3    -1.0    -0.9    -0.8
## log_lik[1195]    -2.2     0.2  0.9    -3.9    -2.8    -2.0    -1.6    -0.8
## log_lik[1196]    -1.0     0.0  0.2    -1.6    -1.0    -0.9    -0.8    -0.8
## log_lik[1197]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1198]    -2.1     0.2  0.6    -3.0    -2.3    -2.1    -1.7    -1.1
## log_lik[1199]    -1.0     0.0  0.2    -1.3    -1.1    -1.0    -0.9    -0.8
## log_lik[1200]    -0.8     0.0  0.0    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[1201]    -1.1     0.1  0.2    -1.5    -1.2    -1.0    -0.9    -0.8
## log_lik[1202]    -1.1     0.0  0.2    -1.4    -1.3    -1.1    -1.0    -0.9
## log_lik[1203]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1204]    -1.9     0.1  0.7    -3.4    -2.4    -1.8    -1.3    -1.0
## log_lik[1205]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1206]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1207]    -0.9     0.0  0.1    -1.3    -0.9    -0.9    -0.8    -0.8
## log_lik[1208]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1209]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1210]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1211]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1212]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1213]    -1.0     0.0  0.2    -1.7    -1.1    -0.9    -0.9    -0.8
## log_lik[1214]    -1.1     0.0  0.2    -1.4    -1.2    -1.0    -0.9    -0.8
## log_lik[1215]    -0.9     0.0  0.1    -1.3    -1.0    -0.8    -0.8    -0.8
## log_lik[1216]    -2.4     0.1  0.6    -3.5    -2.7    -2.4    -2.1    -1.2
## log_lik[1217]    -1.4     0.0  0.3    -1.8    -1.6    -1.5    -1.2    -1.0
## log_lik[1218]    -0.9     0.0  0.1    -1.3    -0.9    -0.9    -0.8    -0.8
## log_lik[1219]    -3.2     0.2  0.8    -4.7    -3.8    -3.4    -2.7    -1.7
## log_lik[1220]    -1.3     0.0  0.3    -1.8    -1.5    -1.3    -1.1    -0.9
## log_lik[1221]    -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[1222]    -4.0     0.1  1.0    -5.4    -4.5    -4.1    -3.2    -2.3
## log_lik[1223]    -1.3     0.0  0.3    -2.1    -1.4    -1.2    -1.1    -1.0
## log_lik[1224]    -0.9     0.0  0.1    -1.0    -0.9    -0.9    -0.8    -0.8
## log_lik[1225]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1226]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1227]    -1.0     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[1228]    -1.2     0.0  0.2    -1.7    -1.4    -1.2    -1.1    -0.8
## log_lik[1229]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1230]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1231]    -1.1     0.0  0.3    -1.6    -1.2    -1.0    -0.9    -0.8
## log_lik[1232]    -1.0     0.0  0.2    -1.6    -1.1    -1.0    -0.9    -0.8
## log_lik[1233]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1234]    -1.4     0.0  0.4    -2.3    -1.6    -1.3    -1.0    -0.8
## log_lik[1235]    -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[1236]    -0.9     0.0  0.2    -1.5    -0.9    -0.8    -0.8    -0.8
## log_lik[1237]    -2.7     0.1  0.7    -4.0    -3.3    -2.6    -2.3    -1.8
## log_lik[1238]    -0.8     0.0  0.1    -1.1    -0.8    -0.8    -0.8    -0.8
## log_lik[1239]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1240]    -1.8     0.1  0.6    -2.7    -2.2    -1.7    -1.4    -1.0
## log_lik[1241]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1242]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1243]    -1.0     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[1244]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1245]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1246]    -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[1247]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1248]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1249]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1250]    -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[1251]    -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1252]    -1.6     0.1  0.6    -2.8    -2.1    -1.5    -1.3    -0.9
## log_lik[1253]    -1.0     0.0  0.2    -1.4    -1.1    -0.9    -0.8    -0.8
## log_lik[1254]    -0.9     0.0  0.2    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1255]    -4.8     0.1  1.1    -6.6    -5.5    -4.9    -3.8    -3.0
## log_lik[1256]    -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[1257]    -0.9     0.0  0.1    -1.2    -1.0    -0.8    -0.8    -0.8
## log_lik[1258]    -3.3     0.1  0.8    -4.7    -3.7    -3.3    -2.9    -1.9
## log_lik[1259]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1260]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1261]    -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[1262]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1263]    -1.1     0.0  0.2    -1.4    -1.2    -1.0    -0.9    -0.9
## log_lik[1264]    -1.0     0.0  0.2    -1.7    -1.2    -0.9    -0.8    -0.8
## log_lik[1265]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1266]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1267]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1268]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1269]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1270]    -1.0     0.0  0.2    -1.4    -1.1    -0.9    -0.9    -0.8
## log_lik[1271]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1272]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1273]    -1.5     0.0  0.3    -1.9    -1.7    -1.5    -1.3    -0.9
## log_lik[1274]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1275]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1276]    -1.0     0.0  0.1    -1.3    -1.1    -1.0    -0.9    -0.8
## log_lik[1277]    -0.8     0.0  0.1    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[1278]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1279]    -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[1280]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1281]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1282]    -0.9     0.0  0.2    -1.4    -0.9    -0.9    -0.8    -0.8
## log_lik[1283]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1284]    -0.9     0.0  0.1    -1.1    -0.8    -0.8    -0.8    -0.8
## log_lik[1285]    -1.2     0.0  0.3    -1.8    -1.2    -1.1    -1.0    -0.8
## log_lik[1286]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1287]    -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1288]    -1.0     0.0  0.3    -1.8    -1.1    -0.9    -0.8    -0.8
## log_lik[1289]    -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[1290]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1291]    -1.0     0.0  0.2    -1.7    -1.0    -0.9    -0.8    -0.8
## log_lik[1292]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1293]    -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[1294]    -1.6     0.1  0.5    -3.0    -1.8    -1.5    -1.2    -1.0
## log_lik[1295]    -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1296]    -1.4     0.0  0.3    -2.1    -1.5    -1.3    -1.2    -0.9
## log_lik[1297]    -1.2     0.0  0.3    -1.8    -1.4    -1.1    -0.9    -0.8
## log_lik[1298]    -0.9     0.0  0.1    -1.2    -1.0    -0.8    -0.8    -0.8
## log_lik[1299]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1300]    -1.0     0.0  0.2    -1.3    -1.0    -0.9    -0.9    -0.8
## log_lik[1301]    -1.0     0.0  0.1    -1.4    -1.0    -0.9    -0.9    -0.8
## log_lik[1302]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1303]    -1.2     0.0  0.3    -1.8    -1.4    -1.1    -0.9    -0.8
## log_lik[1304]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1305]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1306]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1307]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1308]    -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[1309]    -1.2     0.0  0.2    -1.6    -1.3    -1.2    -1.0    -0.8
## log_lik[1310]    -1.0     0.0  0.1    -1.2    -1.0    -0.9    -0.9    -0.8
## log_lik[1311]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1312]    -1.1     0.0  0.3    -1.8    -1.2    -1.0    -0.9    -0.8
## log_lik[1313]    -1.2     0.0  0.3    -1.8    -1.4    -1.3    -1.0    -0.9
## log_lik[1314]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1315]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1316]    -0.9     0.0  0.1    -1.3    -0.9    -0.9    -0.8    -0.8
## log_lik[1317]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1318]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.9    -0.8
## log_lik[1319]    -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1320]    -0.8     0.0  0.1    -0.9    -0.9    -0.8    -0.8    -0.8
## log_lik[1321]    -1.0     0.0  0.2    -1.5    -1.1    -0.9    -0.9    -0.8
## log_lik[1322]    -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1323]    -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[1324]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1325]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1326]    -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1327]    -0.9     0.0  0.2    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[1328]    -1.0     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[1329]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1330]    -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[1331]    -1.2     0.0  0.3    -1.7    -1.4    -1.1    -1.0    -0.8
## log_lik[1332]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1333]    -4.2     0.1  1.1    -6.7    -4.8    -4.1    -3.4    -2.5
## log_lik[1334]    -1.2     0.0  0.3    -1.9    -1.4    -1.2    -1.1    -0.8
## log_lik[1335]    -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[1336]    -3.3     0.1  1.1    -5.2    -4.1    -3.0    -2.5    -1.5
## log_lik[1337]    -1.3     0.0  0.4    -2.1    -1.4    -1.2    -1.0    -0.8
## log_lik[1338]    -1.0     0.0  0.2    -1.7    -1.0    -0.9    -0.8    -0.8
## log_lik[1339]    -3.2     0.1  0.7    -4.5    -3.5    -3.1    -2.7    -2.2
## log_lik[1340]    -1.0     0.0  0.1    -1.3    -1.0    -1.0    -0.9    -0.8
## log_lik[1341]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1342]    -1.7     0.1  0.4    -2.4    -1.9    -1.7    -1.4    -1.0
## log_lik[1343]    -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[1344]    -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[1345]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1346]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1347]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1348]    -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[1349]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1350]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1351]    -1.1     0.0  0.2    -1.7    -1.2    -1.0    -0.9    -0.8
## log_lik[1352]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1353]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1354]    -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[1355]    -1.0     0.0  0.1    -1.3    -1.0    -0.9    -0.9    -0.8
## log_lik[1356]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1357]    -1.3     0.1  0.4    -2.1    -1.5    -1.3    -1.0    -0.8
## log_lik[1358]    -1.2     0.0  0.3    -1.6    -1.3    -1.1    -1.0    -0.8
## log_lik[1359]    -1.0     0.0  0.2    -1.7    -1.1    -0.9    -0.9    -0.8
## log_lik[1360]    -3.3     0.1  1.0    -5.0    -4.0    -3.2    -2.7    -1.6
## log_lik[1361]    -1.1     0.0  0.2    -1.4    -1.3    -1.0    -0.9    -0.8
## log_lik[1362]    -1.1     0.0  0.2    -1.6    -1.2    -1.0    -0.9    -0.8
## log_lik[1363]    -3.3     0.1  0.7    -4.3    -3.8    -3.2    -2.8    -1.8
## log_lik[1364]    -1.2     0.0  0.2    -1.7    -1.3    -1.2    -1.0    -0.9
## log_lik[1365]    -1.0     0.0  0.2    -1.3    -1.1    -1.0    -0.9    -0.8
## log_lik[1366]    -1.4     0.1  0.4    -2.5    -1.5    -1.2    -1.1    -0.9
## log_lik[1367]    -1.1     0.0  0.2    -1.6    -1.2    -1.1    -0.9    -0.8
## log_lik[1368]    -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[1369]    -1.2     0.1  0.5    -2.7    -1.3    -1.1    -0.9    -0.8
## log_lik[1370]    -1.1     0.0  0.2    -1.7    -1.3    -1.0    -1.0    -0.8
## log_lik[1371]    -1.4     0.0  0.4    -2.2    -1.7    -1.3    -1.2    -0.8
## log_lik[1372]    -1.7     0.1  0.4    -2.5    -2.0    -1.7    -1.4    -1.1
## log_lik[1373]    -1.0     0.0  0.2    -1.3    -1.1    -0.9    -0.8    -0.8
## log_lik[1374]    -1.3     0.0  0.3    -1.7    -1.4    -1.2    -1.1    -0.9
## log_lik[1375]    -1.4     0.0  0.4    -2.2    -1.6    -1.4    -1.1    -0.9
## log_lik[1376]    -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[1377]    -1.2     0.0  0.3    -1.7    -1.4    -1.1    -1.0    -0.8
## log_lik[1378]    -1.1     0.1  0.3    -2.1    -1.3    -1.0    -0.9    -0.8
## log_lik[1379]    -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[1380]    -1.1     0.0  0.2    -1.5    -1.2    -1.0    -0.9    -0.8
## log_lik[1381]    -1.7     0.1  0.4    -2.6    -2.0    -1.6    -1.4    -1.2
## log_lik[1382]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1383]    -1.0     0.0  0.1    -1.2    -1.1    -1.0    -0.9    -0.8
## log_lik[1384]    -1.1     0.0  0.3    -1.8    -1.2    -1.0    -1.0    -0.8
## log_lik[1385]    -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1386]    -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[1387]    -1.4     0.1  0.3    -1.9    -1.5    -1.3    -1.2    -0.9
## log_lik[1388]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1389]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1390]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1391]    -1.4     0.0  0.3    -2.1    -1.6    -1.4    -1.1    -0.9
## log_lik[1392]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1393]    -1.1     0.0  0.3    -1.8    -1.2    -1.0    -0.9    -0.8
## log_lik[1394]    -1.3     0.0  0.2    -1.8    -1.5    -1.3    -1.1    -1.0
## log_lik[1395]    -1.0     0.0  0.1    -1.2    -1.0    -0.9    -0.9    -0.8
## log_lik[1396]    -5.6     0.2  1.3    -8.7    -6.2    -5.7    -4.9    -3.5
## log_lik[1397]    -1.0     0.0  0.2    -1.4    -1.1    -1.0    -0.9    -0.8
## log_lik[1398]    -1.1     0.0  0.2    -1.6    -1.2    -1.1    -1.0    -0.9
## log_lik[1399]    -5.5     0.2  1.5    -7.5    -6.5    -5.8    -4.5    -3.1
## log_lik[1400]    -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.8    -0.8
## log_lik[1401]    -1.5     0.1  0.4    -2.4    -1.7    -1.4    -1.2    -1.0
## log_lik[1402]    -6.3     0.2  1.4    -8.1    -7.3    -6.7    -5.2    -3.3
## log_lik[1403]    -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[1404]    -1.2     0.0  0.3    -1.9    -1.4    -1.1    -1.0    -0.8
## log_lik[1405]    -1.3     0.0  0.4    -2.2    -1.5    -1.2    -1.0    -0.8
## log_lik[1406]    -1.4     0.0  0.3    -2.1    -1.8    -1.4    -1.1    -1.0
## log_lik[1407]    -1.7     0.1  0.5    -2.9    -1.8    -1.7    -1.4    -1.1
## log_lik[1408]    -2.2     0.1  0.6    -3.4    -2.5    -2.1    -1.9    -1.1
## log_lik[1409]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1410]    -1.3     0.0  0.3    -2.1    -1.4    -1.2    -1.1    -0.9
## log_lik[1411]    -6.4     0.2  1.4    -9.3    -7.5    -6.3    -5.3    -4.8
## log_lik[1412]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1413]    -1.1     0.0  0.2    -1.5    -1.1    -1.0    -0.9    -0.8
## log_lik[1414]    -2.3     0.1  0.6    -3.4    -2.7    -2.2    -1.9    -1.2
## log_lik[1415]    -1.3     0.0  0.3    -1.9    -1.5    -1.3    -1.1    -1.0
## log_lik[1416]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1417]    -0.9     0.0  0.2    -1.4    -1.0    -0.9    -0.8    -0.8
## log_lik[1418]    -1.1     0.0  0.2    -1.4    -1.2    -1.0    -0.9    -0.9
## log_lik[1419]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1420]    -1.1     0.0  0.3    -1.8    -1.3    -1.1    -0.9    -0.8
## log_lik[1421]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1422]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1423]    -0.9     0.0  0.2    -1.5    -1.0    -0.9    -0.8    -0.8
## log_lik[1424]    -0.8     0.0  0.1    -1.1    -0.8    -0.8    -0.8    -0.8
## log_lik[1425]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1426]    -1.5     0.1  0.4    -2.4    -1.8    -1.3    -1.2    -0.8
## log_lik[1427]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1428]    -0.9     0.0  0.1    -1.3    -0.9    -0.8    -0.8    -0.8
## log_lik[1429]    -1.5     0.1  0.5    -2.6    -1.7    -1.3    -1.1    -0.9
## log_lik[1430]    -1.0     0.0  0.2    -1.5    -1.1    -1.0    -0.9    -0.8
## log_lik[1431]    -0.9     0.0  0.1    -1.1    -0.9    -0.9    -0.8    -0.8
## log_lik[1432]    -2.5     0.1  0.8    -4.2    -2.8    -2.4    -1.8    -1.4
## log_lik[1433]    -1.0     0.0  0.2    -1.5    -1.1    -0.9    -0.9    -0.8
## log_lik[1434]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1435]    -2.3     0.1  0.7    -3.5    -2.9    -2.3    -1.9    -1.3
## log_lik[1436]    -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[1437]    -1.0     0.0  0.2    -1.4    -1.1    -0.9    -0.8    -0.8
## log_lik[1438]    -3.0     0.1  1.1    -5.0    -3.8    -3.0    -1.9    -1.5
## log_lik[1439]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1440]    -0.9     0.0  0.2    -1.4    -1.0    -0.8    -0.8    -0.8
## log_lik[1441]    -1.1     0.0  0.2    -1.6    -1.2    -1.0    -0.9    -0.8
## log_lik[1442]    -1.1     0.0  0.2    -1.4    -1.2    -1.0    -0.9    -0.8
## log_lik[1443]    -0.9     0.0  0.1    -1.1    -1.0    -0.9    -0.8    -0.8
## log_lik[1444]    -1.1     0.1  0.4    -2.1    -1.3    -1.0    -0.9    -0.8
## log_lik[1445]    -1.0     0.0  0.2    -1.5    -1.1    -0.9    -0.9    -0.8
## log_lik[1446]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1447]    -0.9     0.0  0.2    -1.2    -1.1    -0.9    -0.8    -0.8
## log_lik[1448]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1449]    -0.9     0.0  0.1    -1.2    -0.9    -0.9    -0.8    -0.8
## log_lik[1450]    -1.4     0.1  0.4    -2.3    -1.5    -1.3    -1.1    -0.9
## log_lik[1451]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1452]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1453]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1454]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1455]    -0.8     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1456]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1457]    -0.8     0.0  0.1    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1458]    -0.8     0.0  0.1    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1459]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1460]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1461]    -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1462]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1463]    -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1464]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1465]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1466]    -0.8     0.0  0.0    -0.9    -0.8    -0.8    -0.8    -0.8
## log_lik[1467]    -0.8     0.0  0.0    -1.0    -0.8    -0.8    -0.8    -0.8
## log_lik[1468]    -0.9     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1469]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1470]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1471]    -0.9     0.0  0.1    -1.2    -0.9    -0.8    -0.8    -0.8
## log_lik[1472]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1473]    -0.9     0.0  0.1    -1.1    -0.9    -0.8    -0.8    -0.8
## log_lik[1474]    -0.9     0.0  0.1    -1.2    -1.0    -0.9    -0.8    -0.8
## log_lik[1475]    -0.8     0.0  0.1    -1.0    -0.9    -0.8    -0.8    -0.8
## log_lik[1476]    -0.9     0.0  0.1    -1.3    -1.0    -0.9    -0.8    -0.8
## psi[1]            2.0     0.1  1.3     0.3     1.1     1.9     2.6     4.8
## lp__          -2159.2     6.3 21.4 -2195.3 -2174.2 -2158.7 -2150.5 -2108.9
##               n_eff Rhat
## x[1,1]           31  1.0
## x[1,2]           62  1.0
## x[1,3]           43  1.0
## x[1,4]           73  1.0
## x[1,5]           59  1.0
## x[1,6]           60  1.0
## x[1,7]         1672  1.0
## x[1,8]           33  1.0
## x[1,9]           85  1.0
## x[1,10]          76  1.0
## x[1,11]          36  1.0
## x[1,12]          49  1.0
## x[1,13]          52  1.0
## x[1,14]          75  1.0
## x[1,15]         111  1.0
## x[1,16]          24  1.2
## x[1,17]          59  1.1
## x[1,18]          37  1.0
## x[1,19]          56  1.0
## x[1,20]          47  1.0
## x[1,21]          39  1.0
## x[1,22]          99  1.0
## x[1,23]         116  1.0
## x[1,24]          49  1.0
## x[1,25]          56  1.1
## x[1,26]          79  1.0
## x[1,27]          54  1.0
## x[1,28]          49  1.0
## x[1,29]          46  1.0
## x[1,30]          74  1.0
## x[1,31]          56  1.0
## x[1,32]          83  1.0
## x[1,33]          39  1.0
## x[1,34]          86  1.0
## x[1,35]          34  1.0
## x[1,36]          45  1.0
## x[1,37]          32  1.0
## x[1,38]          48  1.0
## x[1,39]          42  1.0
## x[1,40]          31  1.1
## x[1,41]          68  1.0
## x[1,42]          84  1.0
## x[1,43]          50  1.0
## x[1,44]          36  1.0
## x[1,45]          52  1.0
## x[1,46]          30  1.0
## x[1,47]         106  1.0
## x[1,48]          46  1.0
## x[1,49]          31  1.0
## x[1,50]          57  1.0
## x[1,51]          21  1.0
## x[1,52]          67  1.0
## x[1,53]          28  1.0
## x[1,54]          35  1.1
## x[1,55]          51  1.0
## x[1,56]          44  1.0
## x[1,57]          33  1.0
## x[1,58]          35  1.0
## x[1,59]          29  1.0
## x[1,60]          22  1.0
## x[1,61]          17  1.0
## x[1,62]          58  1.0
## x[1,63]       -3762  1.0
## x[1,64]          57  1.0
## x[1,65]          95  1.1
## x[1,66]          25  1.0
## x[1,67]          99  1.1
## x[1,68]          31  1.1
## x[1,69]          60  1.0
## x[1,70]          84  1.0
## x[1,71]         113  1.0
## x[1,72]          53  1.0
## x[1,73]          81  1.0
## x[1,74]          78  1.0
## x[1,75]         118  1.0
## x[1,76]          63  1.0
## x[1,77]          60  1.1
## x[1,78]          52  1.1
## x[1,79]          28  1.2
## x[1,80]          30  1.2
## x[1,81]          95  1.0
## x[1,82]          96  1.0
## x[1,83]         125  1.0
## x[1,84]          59  1.0
## x[1,85]          37  1.0
## x[1,86]          33  1.0
## x[1,87]          46  1.0
## x[1,88]          46  1.0
## x[1,89]          48  1.0
## x[1,90]          72  1.0
## x[1,91]          50  1.0
## x[1,92]          87  1.0
## x[1,93]          76  1.0
## x[1,94]         132  1.0
## x[1,95]          83  1.0
## x[1,96]          82  1.0
## x[1,97]          70  1.0
## x[1,98]          75  1.0
## x[1,99]         365  1.0
## x[1,100]         60  1.0
## x[1,101]         19  1.1
## x[1,102]         59  1.0
## x[1,103]        137  1.0
## x[1,104]         92  1.0
## x[1,105]         96  1.0
## x[1,106]         51  1.0
## x[1,107]         30  1.0
## x[1,108]         36  1.1
## x[1,109]         62  1.0
## x[1,110]         90  1.0
## x[1,111]         29  1.1
## x[1,112]         63  1.0
## x[1,113]         43  1.0
## x[1,114]         40  1.0
## x[1,115]         70  1.0
## x[1,116]         43  1.0
## x[1,117]         64  1.0
## x[1,118]         48  1.0
## x[1,119]         56  1.0
## x[1,120]         43  1.0
## x[1,121]         40  1.0
## x[1,122]         52  1.0
## x[1,123]         54  1.0
## x[1,124]         35  1.0
## x[1,125]         54  1.0
## x[1,126]         53  1.0
## x[1,127]         65  1.0
## x[1,128]      -1763  1.0
## x[1,129]         56  1.0
## x[1,130]         37  1.0
## x[1,131]        105  1.0
## x[1,132]        174  1.0
## x[1,133]         92  1.0
## x[1,134]         44  1.0
## x[1,135]         35  1.0
## x[1,136]         43  1.0
## x[1,137]         48  1.0
## x[1,138]         51  1.0
## x[1,139]         69  1.0
## x[1,140]         34  1.1
## x[1,141]         49  1.0
## x[1,142]         63  1.0
## x[1,143]         84  1.0
## x[1,144]         62  1.0
## x[1,145]         39  1.0
## x[1,146]         50  1.0
## x[1,147]         53  1.0
## x[1,148]         55  1.0
## x[1,149]        138  1.0
## x[1,150]         78  1.0
## x[1,151]         45  1.0
## x[1,152]         43  1.0
## x[1,153]         50  1.0
## x[1,154]         28  1.0
## x[1,155]         51  1.0
## x[1,156]         90  1.0
## x[1,157]        107  1.0
## x[1,158]         36  1.0
## x[1,159]         63  1.0
## x[1,160]         42  1.0
## x[1,161]         64  1.0
## x[1,162]        120  1.0
## x[1,163]         68  1.0
## x[1,164]        157  1.0
## x[1,165]         32  1.0
## x[1,166]        106  1.0
## x[1,167]         68  1.0
## x[1,168]        110  1.0
## x[1,169]        267  1.0
## x[1,170]         66  1.0
## x[1,171]        155  1.0
## x[1,172]         62  1.0
## x[1,173]         86  1.0
## x[1,174]         61  1.0
## x[1,175]         67  1.0
## x[1,176]         88  1.0
## x[1,177]         92  1.0
## x[1,178]         82  1.0
## x[1,179]         62  1.0
## x[1,180]         50  1.0
## x[1,181]         69  1.0
## x[1,182]         49  1.0
## x[1,183]         35  1.0
## x[1,184]         42  1.0
## x[1,185]         38  1.0
## x[1,186]         57  1.0
## x[1,187]         54  1.0
## x[1,188]        107  1.0
## x[1,189]         91  1.0
## x[1,190]         59  1.0
## x[1,191]         59  1.0
## x[1,192]         91  1.0
## x[1,193]         84  1.0
## x[1,194]         87  1.0
## x[1,195]         93  1.0
## x[1,196]         58  1.0
## x[1,197]        214  1.0
## x[1,198]        131  1.0
## x[1,199]        168  1.0
## x[1,200]         74  1.0
## x[1,201]         81  1.0
## x[1,202]         50  1.0
## x[1,203]         52  1.0
## x[1,204]        103  1.0
## x[1,205]        115  1.0
## x[1,206]         77  1.0
## x[1,207]        149  1.0
## x[1,208]        112  1.0
## x[1,209]        101  1.0
## x[1,210]         86  1.0
## x[1,211]         71  1.0
## x[1,212]         51  1.0
## x[1,213]        102  1.0
## x[1,214]         44  1.0
## x[1,215]         61  1.0
## x[1,216]         62  1.0
## x[1,217]         62  1.0
## x[1,218]         34  1.0
## x[1,219]         72  1.0
## x[1,220]         46  1.0
## x[1,221]         44  1.0
## x[1,222]        106  1.0
## x[1,223]         77  1.0
## x[1,224]       1036  1.0
## x[1,225]      12074  1.0
## x[1,226]        119  1.0
## x[1,227]         73  1.0
## x[1,228]        203  1.0
## x[1,229]         43  1.0
## x[1,230]         43  1.0
## x[1,231]        199  1.0
## x[1,232]        175  1.0
## x[1,233]         41  1.0
## x[1,234]         92  1.0
## x[1,235]         91  1.0
## x[1,236]         62  1.0
## x[1,237]         38  1.0
## x[1,238]         37  1.0
## x[1,239]         24  1.0
## x[1,240]         55  1.0
## x[1,241]         56  1.0
## x[1,242]         32  1.0
## x[1,243]         70  1.0
## x[1,244]         43  1.1
## x[1,245]         38  1.0
## x[1,246]         46  1.0
## x[1,247]         87  1.0
## x[1,248]         55  1.0
## x[1,249]         40  1.0
## x[1,250]         39  1.0
## x[1,251]         34  1.0
## x[1,252]         21  1.1
## x[1,253]         63  1.0
## x[1,254]         41  1.0
## x[1,255]         44  1.0
## x[1,256]        313  1.0
## x[1,257]         49  1.0
## x[1,258]         54  1.0
## x[1,259]         63  1.0
## x[1,260]         72  1.0
## x[1,261]        131  1.0
## x[1,262]         95  1.0
## x[1,263]         34  1.0
## x[1,264]         60  1.0
## x[1,265]         79  1.0
## x[1,266]         69  1.0
## x[1,267]       1998  1.0
## x[1,268]        156  1.0
## x[1,269]         53  1.0
## x[1,270]         38  1.0
## x[1,271]         41  1.0
## x[1,272]         62  1.0
## x[1,273]         47  1.0
## x[1,274]         42  1.0
## x[1,275]         49  1.0
## x[1,276]         42  1.0
## x[1,277]         45  1.0
## x[1,278]         57  1.0
## x[1,279]         97  1.0
## x[1,280]         31  1.0
## x[1,281]         81  1.0
## x[1,282]         47  1.0
## x[1,283]         30  1.0
## x[1,284]         33  1.0
## x[1,285]         63  1.0
## x[1,286]        136  1.0
## x[1,287]         30  1.0
## x[1,288]         23  1.0
## x[1,289]         49  1.0
## x[1,290]        102  1.0
## x[1,291]         83  1.0
## x[1,292]         86  1.0
## x[1,293]        128  1.0
## x[1,294]         33  1.0
## x[1,295]         34  1.0
## x[1,296]         27  1.0
## x[1,297]         39  1.0
## x[1,298]         59  1.0
## x[1,299]         85  1.0
## x[1,300]         84  1.0
## x[1,301]        104  1.0
## x[1,302]        141  1.0
## x[1,303]        109  1.0
## x[1,304]         83  1.0
## x[1,305]        115  1.0
## x[1,306]        101  1.0
## x[1,307]         93  1.0
## x[1,308]       -136  1.0
## x[1,309]        153  1.0
## x[1,310]        331  1.0
## x[1,311]         49  1.0
## x[1,312]         69  1.0
## x[1,313]         66  1.0
## x[1,314]         36  1.0
## x[1,315]         62  1.0
## x[1,316]        109  1.0
## x[1,317]        163  1.0
## x[1,318]         54  1.0
## x[1,319]         42  1.0
## x[1,320]         41  1.0
## x[1,321]         63  1.0
## x[1,322]         65  1.0
## x[1,323]        100  1.0
## x[1,324]         59  1.0
## x[1,325]         72  1.0
## x[1,326]         83  1.0
## x[1,327]         61  1.0
## x[1,328]         48  1.0
## x[1,329]         63  1.0
## x[1,330]        167  1.0
## x[1,331]        213  1.0
## x[1,332]         66  1.0
## x[1,333]         29  1.0
## x[1,334]         48  1.0
## x[1,335]         67  1.0
## x[1,336]         53  1.0
## x[1,337]         66  1.0
## x[1,338]        134  1.0
## x[1,339]         93  1.0
## x[1,340]        179  1.0
## x[1,341]        320  1.0
## x[1,342]         44  1.1
## x[1,343]         63  1.0
## x[1,344]         64  1.0
## x[1,345]         87  1.0
## x[1,346]        101  1.0
## x[1,347]         97  1.0
## x[1,348]         85  1.0
## x[1,349]         49  1.0
## x[1,350]        134  1.0
## x[1,351]        101  1.0
## x[1,352]        132  1.0
## x[1,353]       8750  1.0
## x[1,354]         53  1.0
## x[1,355]         92  1.0
## x[1,356]         33  1.0
## x[1,357]         87  1.0
## x[1,358]         73  1.0
## x[1,359]         52  1.0
## x[1,360]         43  1.0
## x[1,361]         45  1.0
## x[1,362]        157  1.0
## x[1,363]       -312  1.0
## x[1,364]         74  1.0
## x[1,365]        114  1.0
## x[1,366]         40  1.0
## x[1,367]        102  1.0
## x[1,368]       -415  1.0
## x[1,369]         47  1.1
## x[1,370]         22  1.1
## x[1,371]        114  1.0
## x[1,372]        105  1.0
## x[1,373]         61  1.0
## x[1,374]         54  1.0
## x[1,375]        153  1.0
## x[1,376]         41  1.0
## x[1,377]         38  1.0
## x[1,378]         96  1.0
## x[1,379]        130  1.0
## x[1,380]        143  1.0
## x[1,381]         60  1.0
## x[1,382]         42  1.0
## x[1,383]         39  1.0
## x[1,384]         77  1.0
## x[1,385]        234  1.0
## x[1,386]         55  1.0
## x[1,387]        468  1.0
## x[1,388]        126  1.0
## x[1,389]         54  1.0
## x[1,390]         84  1.0
## x[1,391]        110  1.0
## x[1,392]        259  1.0
## x[1,393]       -337  1.0
## x[1,394]         44  1.0
## x[1,395]         69  1.0
## x[1,396]         47  1.0
## x[1,397]         32  1.0
## x[1,398]         49  1.0
## x[1,399]         51  1.0
## x[1,400]         30  1.1
## x[1,401]         12  1.2
## x[1,402]         41  1.0
## x[1,403]         94  1.0
## x[1,404]         90  1.0
## x[1,405]        174  1.0
## x[1,406]         48  1.0
## x[1,407]         36  1.0
## x[1,408]         54  1.0
## x[1,409]         53  1.0
## x[1,410]         50  1.0
## x[1,411]         67  1.0
## x[1,412]         84  1.0
## x[1,413]         90  1.0
## x[1,414]        155  1.0
## x[1,415]         77  1.0
## x[1,416]         43  1.0
## x[1,417]         55  1.0
## x[1,418]         74  1.0
## x[1,419]         69  1.0
## x[1,420]         90  1.0
## x[1,421]         58  1.0
## x[1,422]        648  1.0
## x[1,423]        251  1.0
## x[1,424]         94  1.0
## x[1,425]         36  1.0
## x[1,426]         65  1.0
## x[1,427]        116  1.0
## x[1,428]         86  1.0
## x[1,429]        159  1.0
## x[1,430]        124  1.0
## x[1,431]         83  1.0
## x[1,432]         50  1.0
## x[1,433]         37  1.0
## x[1,434]         43  1.1
## x[1,435]         69  1.0
## x[1,436]        133  1.0
## x[1,437]         43  1.1
## x[1,438]         94  1.0
## x[1,439]         75  1.0
## x[1,440]         61  1.0
## x[1,441]         79  1.0
## x[1,442]         46  1.0
## x[1,443]        130  1.0
## x[1,444]       -477  1.0
## x[1,445]        138  1.0
## x[1,446]        125  1.0
## x[1,447]         54  1.0
## x[1,448]        100  1.0
## x[1,449]         62  1.0
## x[1,450]         53  1.0
## x[1,451]         96  1.0
## x[1,452]         91  1.0
## x[1,453]        145  1.0
## x[1,454]        200  1.0
## x[1,455]         91  1.0
## x[1,456]         53  1.0
## x[1,457]         43  1.0
## x[1,458]        141  1.0
## x[1,459]        196  1.0
## x[1,460]         66  1.0
## x[1,461]         66  1.0
## x[1,462]        177  1.0
## x[1,463]         46  1.0
## x[1,464]        106  1.0
## x[1,465]         62  1.0
## x[1,466]         73  1.0
## x[1,467]         98  1.0
## x[1,468]         87  1.0
## x[1,469]         99  1.0
## x[1,470]         73  1.0
## x[1,471]         51  1.0
## x[1,472]         47  1.0
## x[1,473]         42  1.0
## x[1,474]        375  1.0
## x[1,475]         77  1.0
## x[1,476]         70  1.0
## x[1,477]         67  1.0
## x[1,478]         53  1.0
## x[1,479]         79  1.0
## x[1,480]         78  1.0
## x[1,481]        210  1.0
## x[1,482]         81  1.0
## x[1,483]         93  1.0
## x[1,484]         34  1.0
## x[1,485]        506  1.0
## x[1,486]         65  1.0
## x[1,487]         81  1.0
## x[1,488]         38  1.0
## x[1,489]         50  1.0
## x[1,490]         38  1.0
## x[1,491]         49  1.0
## x[1,492]         53  1.0
## Z[1,1]           44  1.0
## Z[2,1]           61  1.0
## Z[3,1]           56  1.0
## sigma[1]         42  1.0
## log_lik[1]      108  1.0
## log_lik[2]       18  1.0
## log_lik[3]       52  1.0
## log_lik[4]       44  1.0
## log_lik[5]       38  1.0
## log_lik[6]       38  1.0
## log_lik[7]       50  1.0
## log_lik[8]       48  1.0
## log_lik[9]       28  1.0
## log_lik[10]      84  1.0
## log_lik[11]      29  1.0
## log_lik[12]      34  1.0
## log_lik[13]      54  1.0
## log_lik[14]      38  1.0
## log_lik[15]      47  1.0
## log_lik[16]      33  1.0
## log_lik[17]      54  1.0
## log_lik[18]      39  1.0
## log_lik[19]      83  1.0
## log_lik[20]      37  1.0
## log_lik[21]      38  1.0
## log_lik[22]      35  1.0
## log_lik[23]      30  1.0
## log_lik[24]      31  1.0
## log_lik[25]     212  1.0
## log_lik[26]      33  1.0
## log_lik[27]      46  1.0
## log_lik[28]     214  1.0
## log_lik[29]      69  1.0
## log_lik[30]      93  1.0
## log_lik[31]      50  1.0
## log_lik[32]      35  1.0
## log_lik[33]      64  1.0
## log_lik[34]      49  1.0
## log_lik[35]      45  1.0
## log_lik[36]      55  1.0
## log_lik[37]      47  1.0
## log_lik[38]      18  1.0
## log_lik[39]      25  1.0
## log_lik[40]      56  1.0
## log_lik[41]      45  1.0
## log_lik[42]      54  1.0
## log_lik[43]      19  1.0
## log_lik[44]      17  1.0
## log_lik[45]      30  1.0
## log_lik[46]      32  1.0
## log_lik[47]      33  1.0
## log_lik[48]      41  1.0
## log_lik[49]      38  1.0
## log_lik[50]      36  1.0
## log_lik[51]      43  1.0
## log_lik[52]      53  1.0
## log_lik[53]      25  1.0
## log_lik[54]      37  1.0
## log_lik[55]      47  1.0
## log_lik[56]      27  1.0
## log_lik[57]      31  1.0
## log_lik[58]      21  1.0
## log_lik[59]      24  1.0
## log_lik[60]      24  1.0
## log_lik[61]      31  1.0
## log_lik[62]      25  1.0
## log_lik[63]      51  1.0
## log_lik[64]      48  1.0
## log_lik[65]      17  1.0
## log_lik[66]      29  1.0
## log_lik[67]      33  1.0
## log_lik[68]      16  1.0
## log_lik[69]      21  1.0
## log_lik[70]      54  1.0
## log_lik[71]      33  1.0
## log_lik[72]      36  1.0
## log_lik[73]      61  1.1
## log_lik[74]      57  1.0
## log_lik[75]      55  1.0
## log_lik[76]      96  1.0
## log_lik[77]      61  1.0
## log_lik[78]      53  1.0
## log_lik[79]      75  1.0
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## log_lik[1151]    83  1.0
## log_lik[1152]    66  1.0
## log_lik[1153]   151  1.0
## log_lik[1154]   205  1.0
## log_lik[1155]   211  1.0
## log_lik[1156]    41  1.0
## log_lik[1157]    52  1.0
## log_lik[1158]    42  1.0
## log_lik[1159]   157  1.0
## log_lik[1160]   191  1.0
## log_lik[1161]   302  1.0
## log_lik[1162]    74  1.0
## log_lik[1163]   405  1.0
## log_lik[1164]   151  1.0
## log_lik[1165]    71  1.0
## log_lik[1166]    28  1.0
## log_lik[1167]    51  1.0
## log_lik[1168]   104  1.0
## log_lik[1169]    65  1.0
## log_lik[1170]    61  1.0
## log_lik[1171]   160  1.0
## log_lik[1172]    48  1.0
## log_lik[1173]    70  1.0
## log_lik[1174]   163  1.0
## log_lik[1175]    56  1.0
## log_lik[1176]    85  1.0
## log_lik[1177]    85  1.0
## log_lik[1178] -1780  1.0
## log_lik[1179]  -109  1.0
## log_lik[1180]    48  1.0
## log_lik[1181]    44  1.0
## log_lik[1182]    45  1.0
## log_lik[1183]    67  1.0
## log_lik[1184]    78  1.0
## log_lik[1185]   127  1.0
## log_lik[1186]    41  1.0
## log_lik[1187]    46  1.0
## log_lik[1188]    55  1.0
## log_lik[1189]    31  1.0
## log_lik[1190]    34  1.0
## log_lik[1191]    40  1.0
## log_lik[1192]    23  1.1
## log_lik[1193]    30  1.0
## log_lik[1194]    67  1.1
## log_lik[1195]    28  1.0
## log_lik[1196]    54  1.0
## log_lik[1197]    34  1.0
## log_lik[1198]    12  1.1
## log_lik[1199]    17  1.1
## log_lik[1200]    46  1.0
## log_lik[1201]    17  1.1
## log_lik[1202]    13  1.2
## log_lik[1203]    31  1.1
## log_lik[1204]    31  1.0
## log_lik[1205]    30  1.0
## log_lik[1206]    30  1.0
## log_lik[1207]    29  1.1
## log_lik[1208]    33  1.1
## log_lik[1209]    33  1.1
## log_lik[1210]    46  1.0
## log_lik[1211]    52  1.0
## log_lik[1212]    55  1.0
## log_lik[1213]    83  1.0
## log_lik[1214]    91  1.0
## log_lik[1215]   119  1.0
## log_lik[1216]    33  1.0
## log_lik[1217]    33  1.0
## log_lik[1218]    49  1.0
## log_lik[1219]    27  1.1
## log_lik[1220]    29  1.1
## log_lik[1221]    45  1.0
## log_lik[1222]    44  1.0
## log_lik[1223]    49  1.0
## log_lik[1224]    36  1.0
## log_lik[1225]    35  1.0
## log_lik[1226]    45  1.0
## log_lik[1227]    32  1.0
## log_lik[1228]    49  1.0
## log_lik[1229]    70  1.0
## log_lik[1230]    33  1.0
## log_lik[1231]    67  1.0
## log_lik[1232]   106  1.0
## log_lik[1233]    40  1.0
## log_lik[1234]   867  1.0
## log_lik[1235]    77  1.0
## log_lik[1236]    37  1.0
## log_lik[1237]    70  1.0
## log_lik[1238]    38  1.1
## log_lik[1239]    52  1.0
## log_lik[1240]    93  1.0
## log_lik[1241]    71  1.0
## log_lik[1242]    63  1.0
## log_lik[1243]    77  1.0
## log_lik[1244]    55  1.0
## log_lik[1245]    45  1.0
## log_lik[1246]    41  1.0
## log_lik[1247]    65  1.0
## log_lik[1248]    42  1.0
## log_lik[1249]    44  1.0
## log_lik[1250]    68  1.0
## log_lik[1251]    48  1.0
## log_lik[1252]    79  1.0
## log_lik[1253]    37  1.0
## log_lik[1254]    64  1.0
## log_lik[1255]    75  1.0
## log_lik[1256]    45  1.0
## log_lik[1257]    86  1.0
## log_lik[1258]    72  1.0
## log_lik[1259]    55  1.0
## log_lik[1260]    34  1.0
## log_lik[1261]    99  1.0
## log_lik[1262]   102  1.0
## log_lik[1263]    64  1.0
## log_lik[1264]   126  1.0
## log_lik[1265]    52  1.0
## log_lik[1266]   116  1.0
## log_lik[1267]    43  1.1
## log_lik[1268]    33  1.0
## log_lik[1269]   101  1.0
## log_lik[1270]    61  1.0
## log_lik[1271]    27  1.0
## log_lik[1272]    33  1.0
## log_lik[1273]    45  1.0
## log_lik[1274]    60  1.0
## log_lik[1275]    45  1.0
## log_lik[1276]    44  1.0
## log_lik[1277]    58  1.0
## log_lik[1278]    36  1.0
## log_lik[1279]    47  1.0
## log_lik[1280]   128  1.0
## log_lik[1281]    48  1.0
## log_lik[1282]    55  1.0
## log_lik[1283]    45  1.0
## log_lik[1284]    21  1.0
## log_lik[1285]   114  1.0
## log_lik[1286]    57  1.0
## log_lik[1287]    24  1.0
## log_lik[1288]    67  1.0
## log_lik[1289]   312  1.0
## log_lik[1290]    27  1.0
## log_lik[1291]    70  1.0
## log_lik[1292]    28  1.0
## log_lik[1293]    91  1.0
## log_lik[1294]    28  1.0
## log_lik[1295]    23  1.1
## log_lik[1296]    43  1.0
## log_lik[1297]    95  1.0
## log_lik[1298]    26  1.0
## log_lik[1299]    31  1.0
## log_lik[1300]    91  1.0
## log_lik[1301]    35  1.0
## log_lik[1302]    18  1.0
## log_lik[1303]   105  1.0
## log_lik[1304]    31  1.0
## log_lik[1305]    42  1.0
## log_lik[1306]    62  1.0
## log_lik[1307]   111  1.0
## log_lik[1308]    62  1.0
## log_lik[1309]    68  1.0
## log_lik[1310]    64  1.0
## log_lik[1311]    48  1.0
## log_lik[1312]    92  1.0
## log_lik[1313]    61  1.0
## log_lik[1314]    22  1.0
## log_lik[1315]    31  1.0
## log_lik[1316]    48  1.0
## log_lik[1317]    35  1.0
## log_lik[1318]    37  1.0
## log_lik[1319]    43  1.0
## log_lik[1320]    57  1.0
## log_lik[1321]    28  1.1
## log_lik[1322]    17  1.2
## log_lik[1323]    89  1.0
## log_lik[1324]    45  1.0
## log_lik[1325]    47  1.0
## log_lik[1326]    33  1.0
## log_lik[1327]    37  1.1
## log_lik[1328]    72  1.0
## log_lik[1329]    28  1.1
## log_lik[1330]    20  1.0
## log_lik[1331]   168  1.0
## log_lik[1332]    24  1.0
## log_lik[1333]    60  1.0
## log_lik[1334]    70  1.0
## log_lik[1335]    60  1.0
## log_lik[1336]   153  1.0
## log_lik[1337]   107  1.0
## log_lik[1338]    50  1.0
## log_lik[1339]    37  1.0
## log_lik[1340]    29  1.0
## log_lik[1341]    33  1.0
## log_lik[1342]    44  1.0
## log_lik[1343]    52  1.0
## log_lik[1344]    77  1.0
## log_lik[1345]    60  1.0
## log_lik[1346]    60  1.0
## log_lik[1347]    31  1.0
## log_lik[1348]    57  1.0
## log_lik[1349]    33  1.0
## log_lik[1350]    45  1.0
## log_lik[1351]    52  1.0
## log_lik[1352]    48  1.0
## log_lik[1353]    38  1.0
## log_lik[1354]    27  1.0
## log_lik[1355]    73  1.0
## log_lik[1356]    37  1.0
## log_lik[1357]    51  1.0
## log_lik[1358]    60  1.0
## log_lik[1359]   301  1.0
## log_lik[1360]    68  1.0
## log_lik[1361]   121  1.0
## log_lik[1362]    84  1.0
## log_lik[1363]    31  1.0
## log_lik[1364]   129  1.0
## log_lik[1365]    63  1.0
## log_lik[1366]    38  1.0
## log_lik[1367]   846  1.0
## log_lik[1368]   110  1.0
## log_lik[1369]    31  1.0
## log_lik[1370]   116  1.0
## log_lik[1371]    69  1.0
## log_lik[1372]    37  1.1
## log_lik[1373]   132  1.0
## log_lik[1374]    30  1.1
## log_lik[1375]    67  1.0
## log_lik[1376]    63  1.0
## log_lik[1377]    39  1.1
## log_lik[1378]    41  1.0
## log_lik[1379]    55  1.0
## log_lik[1380]    44  1.0
## log_lik[1381]    27  1.0
## log_lik[1382]    49  1.0
## log_lik[1383]    48  1.0
## log_lik[1384]    63  1.1
## log_lik[1385]    47  1.1
## log_lik[1386]    83  1.0
## log_lik[1387]    33  1.0
## log_lik[1388]    49  1.1
## log_lik[1389]    68  1.0
## log_lik[1390]    39  1.0
## log_lik[1391]   107  1.0
## log_lik[1392]    44  1.0
## log_lik[1393]    52  1.0
## log_lik[1394]   115  1.0
## log_lik[1395]    72  1.0
## log_lik[1396]    59  1.0
## log_lik[1397]    76  1.0
## log_lik[1398]    77  1.0
## log_lik[1399]    53  1.0
## log_lik[1400]    43  1.0
## log_lik[1401]    24  1.0
## log_lik[1402]    70  1.0
## log_lik[1403]    72  1.0
## log_lik[1404]    95  1.0
## log_lik[1405]    63  1.0
## log_lik[1406]   126  1.0
## log_lik[1407]    64  1.0
## log_lik[1408]   120  1.0
## log_lik[1409]    38  1.0
## log_lik[1410]    58  1.0
## log_lik[1411]    35  1.0
## log_lik[1412]    46  1.0
## log_lik[1413]    35  1.0
## log_lik[1414]    48  1.0
## log_lik[1415]   232  1.0
## log_lik[1416]    37  1.0
## log_lik[1417]    55  1.0
## log_lik[1418]    49  1.0
## log_lik[1419]    30  1.0
## log_lik[1420]    85  1.0
## log_lik[1421]    17  1.0
## log_lik[1422]    16  1.1
## log_lik[1423]    35  1.0
## log_lik[1424]    35  1.0
## log_lik[1425]    56  1.0
## log_lik[1426]    38  1.0
## log_lik[1427]    36  1.0
## log_lik[1428]    50  1.0
## log_lik[1429]    41  1.0
## log_lik[1430]    35  1.0
## log_lik[1431]    32  1.1
## log_lik[1432]    44  1.0
## log_lik[1433]    60  1.0
## log_lik[1434]    40  1.0
## log_lik[1435]    64  1.0
## log_lik[1436]    52  1.0
## log_lik[1437]    83  1.0
## log_lik[1438]    55  1.0
## log_lik[1439]    46  1.0
## log_lik[1440]    39  1.0
## log_lik[1441]   133  1.0
## log_lik[1442]   440  1.0
## log_lik[1443]    59  1.0
## log_lik[1444]    50  1.0
## log_lik[1445]    63  1.0
## log_lik[1446]    42  1.0
## log_lik[1447]    51  1.0
## log_lik[1448]    18  1.0
## log_lik[1449]    57  1.0
## log_lik[1450]    34  1.0
## log_lik[1451]    32  1.0
## log_lik[1452]    67  1.0
## log_lik[1453]    44  1.0
## log_lik[1454]    47  1.0
## log_lik[1455]    62  1.0
## log_lik[1456]    30  1.0
## log_lik[1457]    31  1.0
## log_lik[1458]    38  1.0
## log_lik[1459]    26  1.0
## log_lik[1460]    31  1.0
## log_lik[1461]    32  1.0
## log_lik[1462]    25  1.0
## log_lik[1463]    19  1.0
## log_lik[1464]    31  1.0
## log_lik[1465]    21  1.0
## log_lik[1466]    50  1.0
## log_lik[1467]    45  1.0
## log_lik[1468]    33  1.0
## log_lik[1469]    50  1.0
## log_lik[1470]    51  1.0
## log_lik[1471]    45  1.0
## log_lik[1472]    43  1.0
## log_lik[1473]    46  1.0
## log_lik[1474]    45  1.0
## log_lik[1475]    58  1.0
## log_lik[1476]    70  1.0
## psi[1]          107  1.0
## lp__             11  1.2
## 
## For each parameter, n_eff is a crude measure of effective sample size,
## and Rhat is the potential scale reduction factor on split chains (at 
## convergence, Rhat=1).
ls(dfa.posterior)
## [1] "data"             "model"            "monitor"         
## [4] "samples"          "samples_permuted"
summary(dfa.posterior)
##                  Length Class   Mode   
## model                 1 stanfit S4     
## samples_permuted      6 -none-  list   
## samples          148050 -none-  numeric
## monitor           19740 -none-  numeric
## data               1476 -none-  numeric

3.5 Diagrama de dependencia

Diagrama de dependencia del modelo de factores dinamico en medias

Diagrama de dependencia del modelo de factores dinamico en medias

4 Especificacion dinamica en volatilidad

Seguimos considerando que los datos \((\boldsymbol{X}_t)_{t\geq 1}\) son \((p\times 1)\)-dimensionales y que existen \((\boldsymbol{f}_t)_{t\geq 1}\) vectores latentes \((k\times 1)\)-dimensionales, con \(k<<p\), tales que \[\begin{eqnarray} \boldsymbol{X}_t|\boldsymbol{f}_t & \sim & N_p(\boldsymbol{x}|\Lambda \boldsymbol{f}_t,\Sigma_t)\nonumber \\ \end{eqnarray}\] para \(t\geq 1\), donde \(\Lambda\) es una matriz de cargas de dimension \(p\times k\), y \[\Sigma_t=\text{diag}\{\exp(h_{1t}),\ldots,\exp(h_{pt})\},\] es la matriz de volatilidad estocastica, con \[\begin{eqnarray} \boldsymbol{f}_t|\boldsymbol{G}_{t} & \sim & N_k(\boldsymbol{f}|\boldsymbol{0},\boldsymbol{G}_t),\nonumber \end{eqnarray}\] considerando que \[\boldsymbol{G}_t=\text{diag}\{\exp(g_{1t}),\ldots,\exp(g_{kt})\}.\]

La estructura de dependencia entre las \(\Sigma_t\)s y las \(\boldsymbol{G}_t\)s se define separada y localmente como, \[\begin{eqnarray} h_{tj}|h_{(t-1)j} & \sim & N_1(h|\mu_{hj}+\phi_{hj}(h_{(t-1)j}-\mu_{hj}), s_{hj}),\nonumber \end{eqnarray}\] para \(j=1,\ldots,p\), y \[\begin{eqnarray} g_{ti}|g_{(t-1)i} & \sim & N_1(g|\mu_{gi}+\phi_{gi}(h_{(t-1)i}-\mu_{gi}), s_{gi}),\nonumber \end{eqnarray}\] para \(i=1,\ldots,k\), siendo las colecciones \((s_{hj})_{j=1}^p\) y \((s_{gi})_{i=1}^k\) escalares positivos conocidos.

4.1 Parametros

Asi, los parametros del modelo por estimar tornan en \[ \left(\Lambda,(\mu_{hj},\phi_{hj})_{j=1}^{p},(\mu_{gi},\phi_{gi})_{i=1}^{k}\right), \] y las variables latentes son \[ \left((\boldsymbol{f}_t)_{t\geq 1},(\boldsymbol{h}_t)_{t\geq 1},(\boldsymbol{g}_t)_{t\geq 1}\right) \approx \left((\boldsymbol{f}_t)_{t\geq 1},(\boldsymbol{\Sigma}_t)_{t\geq 1},(\boldsymbol{G}_t)_{t\geq 1}\right). \]

4.2 Observacion

Aunque \(\Lambda\) es fijo en \(t\), la estructura de codependencia de la \(p\) variables originales es reproducida por el modelo actual a traves de \(\boldsymbol{G}_t\), porque \[ \Omega_t=\Lambda\boldsymbol{G}_t\Lambda' + \boldsymbol{\Sigma}_t, \] para \(t\geq 1.\)

4.3 Implementacion

if(!require("factorstochvol")){install.packages("factorstochvol")}
## Loading required package: factorstochvol
library("factorstochvol")

datatsc.3 <- as.matrix(datatsc.3)

dsfa.posterior <- fsvsample(datatsc.3, 
                            factors = 1, 
                            draws = 150, burnin = 50, runningstore = 6)
## 
## Calling 1-factor MCMC sampler for 3 series of length 492.
## 
## 
********* Iteration   1 of 200 (  0%) *********
********* Iteration 137 of 200 ( 68%) *********
********* Iteration 200 of 200 (100%) *********
## 
## Reorganizing runningstores... Done!
## Ex-post sign-identification... Done!
voltimeplot(dsfa.posterior)

corimageplot(dsfa.posterior, nrow(datatsc.3), plotCI = 'circle')

oldpar <- par(ask = FALSE)
plot(dsfa.posterior)

par(oldpar)

4.4 Diagrama de dependencia

Diagrama de dependencia del modelo de factores dinamico en volatilidad

Diagrama de dependencia del modelo de factores dinamico en volatilidad

5 Comentarios

5.1 Generales

  • Los modelos flexibles que hemos visto en esta sesion presentan resultados ad-hoc que son confiables.

  • Aunque confiables, estas soluciones no son generales y por lo menos dominantes a todas las circunstancias.

  • Hemos visto que cada implementacion tiene estructuras de codificacion, uso y notacion particulares. Estos aspectos siempre podran sortearse teniendo conocimiento de aspecto metodologico que involucra el modelo.

  • En estas ilustraciones hemos corrido pocas simulaciones de las cadenas de Markov para hacer inferencia… En aplicaciones reales deberan correr mas, no demasiadas, de hecho.

5.2 Adecuacion del modelo

Hemos visto varias especificaciones del modelo de factores:

  • Estatico

  • Dinamico en nivel

  • Dinamico en nivel y volatilidad

  • Entre otras…

Adicionalmente, podemos considerar la capa de especificacion respecto a la reduccion de dimensionalidad.

Es asi pertinente ahora estudiar como cuantificar la adecuacion del modelo ante estas especificaciones.

5.2.1 Especificacion

Grosso modo, tendremos que para un conjunto de datos \[ \text{datos}=(\boldsymbol{x}_1,\ldots,\boldsymbol{x}_T), \] tedremos un conjunto de especificaciones \(m=1,\ldots,M\) (siendo \(M\) el numero todal de especificaciones), para las que podemos calcular la devianza del modelo, \(DIC(m)\), definido como \[ DIC(m)=-2 \widehat{\text{log-lik}}(\text{Modelo }m|\text{datos}) + 2\text{ep}(m), \] donde

  • \(\widehat{\text{log-lik}}(\text{Modelo }m|\text{datos})\) denota la logverosimilitud del modelo \(m\) evaluada en el estimador puntual (frecuentista o bayesiano) de los parametros del modelo \(m\), y

  • \(\text{ep}(m)\) denota el numero de parametros efectivos (medias y covarianzas) del modelo \(m\).

5.2.2 Calculo

A partir de las estimaciones via simulacion, podemos, en general, calcular los estimados puntuales de las medias, \[ \left\{\widehat{\boldsymbol{\mu}}_1,\ldots,\widehat{\boldsymbol{\mu}}_T\right\}, \] y de la estructura de covariabilidad, \[ \left\{\widehat{\boldsymbol{\Lambda}}_1,\ldots,\widehat{\boldsymbol{\Lambda}}_T\right\}, \] y \[ \left\{\widehat{\boldsymbol{\Sigma}}_1,\ldots,\widehat{\boldsymbol{\Sigma}}_T\right\}. \]

De manera general, la logverosimilitud del modelo \(m\) evaluada en los estimadores puntuales es calculada como \[ \widehat{\text{log-lik}}(\text{Modelo }m|\text{datos}) = \sum_{t=1}^{T}\text{log}\left\{ \text{N}_p\left(\boldsymbol{x}_t|\widehat{\boldsymbol{\mu}}_t,\widehat{\boldsymbol{\Omega}}_t\right)\right\}, \] donde \[ \widehat{\boldsymbol{\Omega}}_t =\widehat{\boldsymbol{\Lambda}}_t\widehat{\boldsymbol{\Lambda}}_1'+\widehat{\boldsymbol{\Sigma}}_t, \] para \(t=1,\ldots,T\).

Nota.- Si se trata de un modelo reducido (estructuralmente mas simple), entonces esta logverosimilitud puede simplificarse de acuerdo a tal estructura.

6 Lecturas complementarias

  • Jackson et al (2015) Specification and Estimation of Bayesian Dynamic Factor Models, est46114_s13_suplemento1.pdf

  • Kastner et al (2017) Efficient bayesian Inference for Multivariate Factor Stochastic Volatility Models, est46114_s13_suplemento2.pdf

  • West & Harrison (1997) Bayesian Forecasting and Dinamic Models, Springer, “Cap. Multivariate Modelling and Forecasting”.